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| Meta Title | Differential Equations - Review : Eigenvalues & Eigenvectors |
| Meta Description | In this section we will introduce the concept of eigenvalues and eigenvectors of a matrix. We define the characteristic polynomial and show how it can be used to find the eigenvalues for a matrix. Once we have the eigenvalues for a matrix we also show how to find the corresponding eigenvalues for the matrix. |
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Section 5.3 : Review : Eigenvalues & Eigenvectors
If you get nothing out of this quick review of linear algebra you must get this section. Without this section you will not be able to do any of the differential equations work that is in this chapter.
So, let’s start with the following. If we multiply an
n
×
n
matrix by an
n
×
1
vector we will get a new
n
×
1
vector back. In other words,
A
η
→
=
y
→
What we want to know is if it is possible for the following to happen. Instead of just getting a brand new vector out of the multiplication is it possible instead to get the following,
(1)
A
η
→
=
λ
η
→
In other words, is it possible, at least for certain
λ
and
η
→
, to have matrix multiplication be the same as just multiplying the vector by a constant? Of course, we probably wouldn’t be talking about this if the answer was no. So, it is possible for this to happen, however, it won’t happen for just any value of
λ
or
η
→
. If we do happen to have a
λ
and
η
→
for which this works (and they will always come in pairs) then we call
λ
an
eigenvalue
of
A
and
η
→
an
eigenvector
of
A
.
So, how do we go about finding the eigenvalues and eigenvectors for a matrix? Well first notice that if
η
→
=
0
→
then
(1)
is going to be true for any value of
λ
and so we are going to make the assumption that
η
→
≠
0
→
. With that out of the way let’s rewrite
(1)
a little.
A
η
→
−
λ
η
→
=
0
→
A
η
→
−
λ
I
n
η
→
=
0
→
(
A
−
λ
I
n
)
η
→
=
0
→
Notice that before we factored out the
η
→
we added in the appropriately sized identity matrix. This is equivalent to multiplying things by a one and so doesn’t change the value of anything. We needed to do this because without it we would have had the difference of a matrix,
A
, and a constant,
λ
, and this can’t be done. We now have the difference of two matrices of the same size which can be done.
So, with this rewrite we see that
(2)
(
A
−
λ
I
n
)
η
→
=
0
→
is equivalent to
(1)
. In order to find the eigenvectors for a matrix we will need to solve a homogeneous system. Recall the
fact
from the previous section that we know that we will either have exactly one solution (
η
→
=
0
→
) or we will have infinitely many nonzero solutions. Since we’ve already said that we don’t want
η
→
=
0
→
this means that we want the second case.
Knowing this will allow us to find the eigenvalues for a matrix. Recall from this fact that we will get the second case only if the matrix in the system is singular. Therefore, we will need to determine the values of
λ
for which we get,
det
(
A
−
λ
I
)
=
0
Once we have the eigenvalues we can then go back and determine the eigenvectors for each eigenvalue. Let’s take a look at a couple of quick facts about eigenvalues and eigenvectors.
Fact
If
A
is an
n
×
n
matrix then
det
(
A
−
λ
I
)
=
0
is an
n
th
degree polynomial. This polynomial is called the
characteristic polynomial
.
To find eigenvalues of a matrix all we need to do is solve a polynomial. That’s generally not too bad provided we keep
n
small. Likewise this fact also tells us that for an
n
×
n
matrix,
A
, we will have
n
eigenvalues if we include all repeated eigenvalues.
Fact
If
λ
1
,
λ
2
,
…
,
λ
n
is the complete list of eigenvalues for
A
(including all repeated eigenvalues) then,
If
λ
occurs only once in the list then we call
λ
simple
.
If
λ
occurs
k
>
1
times in the list then we say that
λ
has
multiplicity
k
.
If
λ
1
,
λ
2
,
…
,
λ
k
(
k
≤
n
) are the simple eigenvalues in the list with corresponding eigenvectors
η
→
(
1
)
,
η
→
(
2
)
, …,
η
→
(
k
)
then the eigenvectors are all linearly independent.
If
λ
is an eigenvalue of multiplicity
k
>
1
then
λ
will have anywhere from 1 to
k
linearly independent eigenvectors.
The usefulness of these facts will become apparent when we get back into differential equations since in that work we will want linearly independent solutions.
Let’s work a couple of examples now to see how we actually go about finding eigenvalues and eigenvectors.
Example 1
Find the eigenvalues and eigenvectors of the following matrix.
A
=
(
2
7
−
1
−
6
)
Show Solution
The first thing that we need to do is find the eigenvalues. That means we need the following matrix,
A
−
λ
I
=
(
2
7
−
1
−
6
)
−
λ
(
1
0
0
1
)
=
(
2
−
λ
7
−
1
−
6
−
λ
)
In particular we need to determine where the determinant of this matrix is zero.
det
(
A
−
λ
I
)
=
(
2
−
λ
)
(
−
6
−
λ
)
+
7
=
λ
2
+
4
λ
−
5
=
(
λ
+
5
)
(
λ
−
1
)
So, it looks like we will have two simple eigenvalues for this matrix,
λ
1
=
−
5
and
λ
2
=
1
. We will now need to find the eigenvectors for each of these. Also note that according to the fact above, the two eigenvectors should be linearly independent.
To find the eigenvectors we simply plug in each eigenvalue into and solve. So, let’s do that.
λ
1
=
−
5
:
In this case we need to solve the following system.
(
7
7
−
1
−
1
)
η
→
=
(
0
0
)
Recall that officially to solve this system we use the following augmented matrix.
(
7
7
0
−
1
−
1
0
)
1
7
R
1
+
R
2
⇒
(
7
7
0
0
0
0
)
Upon reducing down we see that we get a single equation
7
η
1
+
7
η
2
=
0
⇒
η
1
=
−
η
2
that will yield an infinite number of solutions. This is expected behavior. Recall that we picked the eigenvalues so that the matrix would be singular and so we would get infinitely many solutions.
Notice as well that we could have identified this from the original system. This won’t always be the case, but in the
2
×
2
case we can see from the system that one row will be a multiple of the other and so we will get infinite solutions. From this point on we won’t be actually solving systems in these cases. We will just go straight to the equation and we can use either of the two rows for this equation.
Now, let’s get back to the eigenvector, since that is what we were after. In general then the eigenvector will be any vector that satisfies the following,
η
→
=
(
η
1
η
2
)
=
(
−
η
2
η
2
)
,
η
2
≠
0
To get this we used the solution to the equation that we found above.
We really don’t want a general eigenvector however so we will pick a value for
η
2
to get a specific eigenvector. We can choose anything (except
η
2
=
0
), so pick something that will make the eigenvector “nice”. Note as well that since we’ve already assumed that the eigenvector is not zero we must choose a value that will not give us zero, which is why we want to avoid
η
2
=
0
in this case. Here’s the eigenvector for this eigenvalue.
η
→
(
1
)
=
(
−
1
1
)
,
using
η
2
=
1
Now we get to do this all over again for the second eigenvalue.
λ
2
=
1
:
We’ll do much less work with this part than we did with the previous part. We will need to solve the following system.
(
1
7
−
1
−
7
)
η
→
=
(
0
0
)
Clearly both rows are multiples of each other and so we will get infinitely many solutions. We can choose to work with either row. We’ll run with the first because to avoid having too many minus signs floating around. Doing this gives us,
η
1
+
7
η
2
=
0
η
1
=
−
7
η
2
Note that we can solve this for either of the two variables. However, with an eye towards working with these later on let’s try to avoid as many fractions as possible. The eigenvector is then,
η
→
=
(
η
1
η
2
)
=
(
−
7
η
2
η
2
)
,
η
2
≠
0
η
→
(
2
)
=
(
−
7
1
)
,
using
η
2
=
1
Summarizing we have,
λ
1
=
−
5
η
→
(
1
)
=
(
−
1
1
)
λ
2
=
1
η
→
(
2
)
=
(
−
7
1
)
Note that the two eigenvectors are linearly independent as predicted.
Example 2
Find the eigenvalues and eigenvectors of the following matrix.
A
=
(
1
−
1
4
9
−
1
3
)
Show Solution
This matrix has fractions in it. That’s life so don’t get excited about it. First, we need the eigenvalues.
det
(
A
−
λ
I
)
=
|
1
−
λ
−
1
4
9
−
1
3
−
λ
|
=
(
1
−
λ
)
(
−
1
3
−
λ
)
+
4
9
=
λ
2
−
2
3
λ
+
1
9
=
(
λ
−
1
3
)
2
⇒
λ
1
,
2
=
1
3
So, it looks like we’ve got an eigenvalue of multiplicity 2 here. Remember that the power on the term will be the multiplicity.
Now, let’s find the eigenvector(s). This one is going to be a little different from the first example. There is only one eigenvalue so let’s do the work for that one. We will need to solve the following system,
(
2
3
−
1
4
9
−
2
3
)
(
η
1
η
2
)
=
(
0
0
)
⇒
R
1
=
3
2
R
2
So, the rows are multiples of each other. We’ll work with the first equation in this example to find the eigenvector.
2
3
η
1
−
η
2
=
0
η
2
=
2
3
η
1
Recall in the last example we decided that we wanted to make these as “nice” as possible and so should avoid fractions if we can. Sometimes, as in this case, we simply can’t so we’ll have to deal with it. In this case the eigenvector will be,
η
→
=
(
η
1
η
2
)
=
(
η
1
2
3
η
1
)
,
η
1
≠
0
η
→
(
1
)
=
(
3
2
)
,
η
1
=
3
Note that by careful choice of the variable in this case we were able to get rid of the fraction that we had. This is something that in general doesn’t much matter if we do or not. However, when we get back to differential equations it will be easier on us if we don’t have any fractions so we will usually try to eliminate them at this step.
Also, in this case we are only going to get a single (linearly independent) eigenvector. We can get other eigenvectors, by choosing different values of
η
1
. However, each of these will be linearly dependent with the first eigenvector. If you’re not convinced of this try it. Pick some values for
η
1
and get a different vector and check to see if the two are linearly dependent.
Recall from the fact above that an eigenvalue of multiplicity
k
will have anywhere from 1 to
k
linearly independent eigenvectors. In this case we got one. For most of the
2
×
2
matrices that we’ll be working with this will be the case, although it doesn’t have to be. We can, on occasion, get two.
Example 3
Find the eigenvalues and eigenvectors of the following matrix.
A
=
(
−
4
−
17
2
2
)
Show Solution
So, we’ll start with the eigenvalues.
det
(
A
−
λ
I
)
=
|
−
4
−
λ
−
17
2
2
−
λ
|
=
(
−
4
−
λ
)
(
2
−
λ
)
+
34
=
λ
2
+
2
λ
+
26
This doesn’t factor, so upon using the quadratic formula we arrive at,
λ
1
,
2
=
−
1
±
5
i
In this case we get complex eigenvalues which are definitely a fact of life with eigenvalue/eigenvector problems so get used to them.
Finding eigenvectors for complex eigenvalues is identical to the previous two examples, but it will be somewhat messier. So, let’s do that.
λ
1
=
−
1
+
5
i
:
The system that we need to solve this time is
(
−
4
−
(
−
1
+
5
i
)
−
17
2
2
−
(
−
1
+
5
i
)
)
(
η
1
η
2
)
=
(
0
0
)
(
−
3
−
5
i
−
17
2
3
−
5
i
)
(
η
1
η
2
)
=
(
0
0
)
Now, it’s not super clear that the rows are multiples of each other, but they are. In this case we have,
R
1
=
−
1
2
(
3
+
5
i
)
R
2
This is not something that you need to worry about, we just wanted to make the point. For the work that we’ll be doing later on with differential equations we will just assume that we’ve done everything correctly and we’ve got two rows that are multiples of each other. Therefore, all that we need to do here is pick one of the rows and work with it.
We’ll work with the second row this time.
2
η
1
+
(
3
−
5
i
)
η
2
=
0
Now we can solve for either of the two variables. However, again looking forward to differential equations, we are going to need the “
i
” in the numerator so solve the equation in such a way as this will happen. Doing this gives,
2
η
1
=
−
(
3
−
5
i
)
η
2
η
1
=
−
1
2
(
3
−
5
i
)
η
2
So, the eigenvector in this case is
η
→
=
(
η
1
η
2
)
=
(
−
1
2
(
3
−
5
i
)
η
2
η
2
)
,
η
2
≠
0
η
→
(
1
)
=
(
−
3
+
5
i
2
)
,
η
2
=
2
As with the previous example we choose the value of the variable to clear out the fraction.
Now, the work for the second eigenvector is almost identical and so we’ll not dwell on that too much.
λ
2
=
−
1
−
5
i
:
The system that we need to solve here is
(
−
4
−
(
−
1
−
5
i
)
−
17
2
2
−
(
−
1
−
5
i
)
)
(
η
1
η
2
)
=
(
0
0
)
(
−
3
+
5
i
−
17
2
3
+
5
i
)
(
η
1
η
2
)
=
(
0
0
)
Working with the second row again gives,
2
η
1
+
(
3
+
5
i
)
η
2
=
0
⇒
η
1
=
−
1
2
(
3
+
5
i
)
η
2
The eigenvector in this case is
η
→
=
(
η
1
η
2
)
=
(
−
1
2
(
3
+
5
i
)
η
2
η
2
)
,
η
2
≠
0
η
→
(
2
)
=
(
−
3
−
5
i
2
)
,
η
2
=
2
Summarizing,
λ
1
=
−
1
+
5
i
η
→
(
1
)
=
(
−
3
+
5
i
2
)
λ
2
=
−
1
−
5
i
η
→
(
2
)
=
(
−
3
−
5
i
2
)
There is a nice fact that we can use to simplify the work when we get complex eigenvalues. We need a bit of terminology first however.
If we start with a complex number,
z
=
a
+
b
i
then the
complex conjugate
of
z
is
z
¯
=
a
−
b
i
To compute the complex conjugate of a complex number we simply change the sign on the term that contains the “
i
”. The complex conjugate of a vector is just the conjugate of each of the vector’s components.
We now have the following fact about complex eigenvalues and eigenvectors.
Fact
If
A
is an
n
×
n
matrix with only real numbers and if
λ
1
=
a
+
b
i
is an eigenvalue with eigenvector
η
→
(
1
)
. Then
λ
2
=
λ
1
¯
=
a
−
b
i
is also an eigenvalue and its eigenvector is the conjugate of
η
→
(
1
)
.
This fact is something that you should feel free to use as you need to in our work.
Now, we need to work one final eigenvalue/eigenvector problem. To this point we’ve only worked with
2
×
2
matrices and we should work at least one that isn’t
2
×
2
. Also, we need to work one in which we get an eigenvalue of multiplicity greater than one that has more than one linearly independent eigenvector.
Example 4
Find the eigenvalues and eigenvectors of the following matrix.
A
=
(
0
1
1
1
0
1
1
1
0
)
Show Solution
Despite the fact that this is a
3
×
3
matrix, it still works the same as the
2
×
2
matrices that we’ve been working with. So, start with the eigenvalues
det
(
A
−
λ
I
)
=
|
−
λ
1
1
1
−
λ
1
1
1
−
λ
|
=
−
λ
3
+
3
λ
+
2
=
(
λ
−
2
)
(
λ
+
1
)
2
λ
1
=
2
,
λ
2
,
3
=
−
1
So, we’ve got a simple eigenvalue and an eigenvalue of multiplicity 2. Note that we used the same method of computing the determinant of a
3
×
3
matrix that we used in the previous section. We just didn’t show the work.
Let’s now get the eigenvectors. We’ll start with the simple eigenvector.
λ
1
=
2
:
Here we’ll need to solve,
(
−
2
1
1
1
−
2
1
1
1
−
2
)
(
η
1
η
2
η
3
)
=
(
0
0
0
)
This time, unlike the
2
×
2
cases we worked earlier, we actually need to solve the system. So let’s do that.
(
−
2
1
1
0
1
−
2
1
0
1
1
−
2
0
)
R
1
↔
R
2
⇒
(
1
−
2
1
0
−
2
1
1
0
1
1
−
2
0
)
R
2
+
2
R
1
R
3
−
R
1
⇒
(
1
−
2
1
0
0
−
3
3
0
0
3
−
3
0
)
−
1
3
R
2
⇒
(
1
−
2
1
0
0
1
−
1
0
0
3
−
3
0
)
R
3
−
3
R
2
R
1
+
2
R
2
⇒
(
1
0
−
1
0
0
1
−
1
0
0
0
0
0
)
Going back to equations gives,
η
1
−
η
3
=
0
⇒
η
1
=
η
3
η
2
−
η
3
=
0
⇒
η
2
=
η
3
So, again we get infinitely many solutions as we should for eigenvectors. The eigenvector is then,
η
→
=
(
η
1
η
2
η
3
)
=
(
η
3
η
3
η
3
)
,
η
3
≠
0
η
→
(
1
)
=
(
1
1
1
)
,
η
3
=
1
Now, let’s do the other eigenvalue.
λ
2
=
−
1
:
Here we’ll need to solve,
(
1
1
1
1
1
1
1
1
1
)
(
η
1
η
2
η
3
)
=
(
0
0
0
)
Okay, in this case is clear that all three rows are the same and so there isn’t any reason to actually solve the system since we can clear out the bottom two rows to all zeroes in one step. The equation that we get then is,
η
1
+
η
2
+
η
3
=
0
⇒
η
1
=
−
η
2
−
η
3
So, in this case we get to pick two of the values for free and will still get infinitely many solutions. Here is the general eigenvector for this case,
η
→
=
(
η
1
η
2
η
3
)
=
(
−
η
2
−
η
3
η
2
η
3
)
,
η
2
≠
0
and
η
3
≠
0
at the same time
Notice the restriction this time. Recall that we only require that the eigenvector not be the zero vector. This means that we can allow one or the other of the two variables to be zero, we just can’t allow both of them to be zero at the same time!
What this means for us is that we are going to get two linearly independent eigenvectors this time. Here they are.
η
→
(
2
)
=
(
−
1
0
1
)
η
2
=
0
and
η
3
=
1
η
→
(
3
)
=
(
−
1
1
0
)
η
2
=
1
and
η
3
=
0
Now when we talked about linear independent vectors in the last
section
we only looked at
n
vectors each with
n
components. We can still talk about linear independence in this case however. Recall
back
with we did linear independence for functions we saw at the time that if two functions were linearly dependent then they were multiples of each other. Well the same thing holds true for vectors. Two vectors will be linearly dependent if they are multiples of each other.
In this case there is no way to get
η
→
(
2
)
by multiplying
η
→
(
3
)
by a constant. Therefore, these two vectors must be linearly independent.
So, summarizing up, here are the eigenvalues and eigenvectors for this matrix
λ
1
=
2
η
→
(
1
)
=
(
1
1
1
)
λ
2
=
−
1
η
→
(
2
)
=
(
−
1
0
1
)
λ
3
=
−
1
η
→
(
3
)
=
(
−
1
1
0
) |
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- [5\.4 More Substitution Rule](https://tutorial.math.lamar.edu/Classes/CalcI/SubstitutionRuleIndefinitePtII.aspx)
- [5\.5 Area Problem](https://tutorial.math.lamar.edu/Classes/CalcI/AreaProblem.aspx)
- [5\.6 Definition of the Definite Integral](https://tutorial.math.lamar.edu/Classes/CalcI/DefnOfDefiniteIntegral.aspx)
- [5\.7 Computing Definite Integrals](https://tutorial.math.lamar.edu/Classes/CalcI/ComputingDefiniteIntegrals.aspx)
- [5\.8 Substitution Rule for Definite Integrals](https://tutorial.math.lamar.edu/Classes/CalcI/SubstitutionRuleDefinite.aspx)
[Close submenu (6. Applications of Integrals)](https://tutorial.math.lamar.edu/classes/de/la_eigen.aspx#mm-10)[6\. Applications of Integrals](https://tutorial.math.lamar.edu/classes/de/la_eigen.aspx#mm-10)[Pauls Notes](https://tutorial.math.lamar.edu/classes/de/la_eigen.aspx#mm-1)/[Calculus I](https://tutorial.math.lamar.edu/classes/de/la_eigen.aspx#mm-10)/6\. Applications of Integrals
- [6\.1 Average Function Value](https://tutorial.math.lamar.edu/Classes/CalcI/AvgFcnValue.aspx)
- [6\.2 Area Between Curves](https://tutorial.math.lamar.edu/Classes/CalcI/AreaBetweenCurves.aspx)
- [6\.3 Volumes of Solids of Revolution / Method of Rings](https://tutorial.math.lamar.edu/Classes/CalcI/VolumeWithRings.aspx)
- [6\.4 Volumes of Solids of Revolution/Method of Cylinders](https://tutorial.math.lamar.edu/Classes/CalcI/VolumeWithCylinder.aspx)
- [6\.5 More Volume Problems](https://tutorial.math.lamar.edu/Classes/CalcI/MoreVolume.aspx)
- [6\.6 Work](https://tutorial.math.lamar.edu/Classes/CalcI/Work.aspx)
[Close submenu (Appendix A. Extras)](https://tutorial.math.lamar.edu/classes/de/la_eigen.aspx#mm-10)[Appendix A. Extras](https://tutorial.math.lamar.edu/classes/de/la_eigen.aspx#mm-10)[Pauls Notes](https://tutorial.math.lamar.edu/classes/de/la_eigen.aspx#mm-1)/[Calculus I](https://tutorial.math.lamar.edu/classes/de/la_eigen.aspx#mm-10)/Appendix A. Extras
- [A.1 Proof of Various Limit Properties](https://tutorial.math.lamar.edu/Classes/CalcI/LimitProofs.aspx)
- [A.2 Proof of Various Derivative Properties](https://tutorial.math.lamar.edu/Classes/CalcI/DerivativeProofs.aspx)
- [A.3 Proof of Trig Limits](https://tutorial.math.lamar.edu/Classes/CalcI/ProofTrigDeriv.aspx)
- [A.4 Proofs of Derivative Applications Facts](https://tutorial.math.lamar.edu/Classes/CalcI/DerivativeAppsProofs.aspx)
- [A.5 Proof of Various Integral Properties](https://tutorial.math.lamar.edu/Classes/CalcI/ProofIntProp.aspx)
- [A.6 Area and Volume Formulas](https://tutorial.math.lamar.edu/Classes/CalcI/Area_Volume_Formulas.aspx)
- [A.7 Types of Infinity](https://tutorial.math.lamar.edu/Classes/CalcI/TypesOfInfinity.aspx)
- [A.8 Summation Notation](https://tutorial.math.lamar.edu/Classes/CalcI/SummationNotation.aspx)
- [A.9 Constant of Integration](https://tutorial.math.lamar.edu/Classes/CalcI/ConstantofIntegration.aspx)
[Close submenu (Calculus II)](https://tutorial.math.lamar.edu/classes/de/la_eigen.aspx#mm-1)[Calculus II](https://tutorial.math.lamar.edu/classes/de/la_eigen.aspx#mm-1)[Pauls Notes](https://tutorial.math.lamar.edu/classes/de/la_eigen.aspx#mm-1)/Calculus II
- [Open submenu (7. Integration Techniques)](https://tutorial.math.lamar.edu/classes/de/la_eigen.aspx#mm-19)[7\. Integration Techniques](https://tutorial.math.lamar.edu/Classes/CalcII/IntTechIntro.aspx)
- [Open submenu (8. Applications of Integrals)](https://tutorial.math.lamar.edu/classes/de/la_eigen.aspx#mm-20)[8\. Applications of Integrals](https://tutorial.math.lamar.edu/Classes/CalcII/IntAppsIntro.aspx)
- [Open submenu (9. Parametric Equations and Polar Coordinates)](https://tutorial.math.lamar.edu/classes/de/la_eigen.aspx#mm-21)[9\. Parametric Equations and Polar Coordinates](https://tutorial.math.lamar.edu/Classes/CalcII/ParametricIntro.aspx)
- [Open submenu (10. Series & Sequences)](https://tutorial.math.lamar.edu/classes/de/la_eigen.aspx#mm-22)[10\. Series & Sequences](https://tutorial.math.lamar.edu/Classes/CalcII/SeriesIntro.aspx)
- [Open submenu (11. Vectors)](https://tutorial.math.lamar.edu/classes/de/la_eigen.aspx#mm-23)[11\. Vectors](https://tutorial.math.lamar.edu/Classes/CalcII/VectorsIntro.aspx)
- [Open submenu (12. 3-Dimensional Space)](https://tutorial.math.lamar.edu/classes/de/la_eigen.aspx#mm-24)[12\. 3-Dimensional Space](https://tutorial.math.lamar.edu/Classes/CalcII/3DSpace.aspx)
[Close submenu (7. Integration Techniques)](https://tutorial.math.lamar.edu/classes/de/la_eigen.aspx#mm-18)[7\. Integration Techniques](https://tutorial.math.lamar.edu/classes/de/la_eigen.aspx#mm-18)[Pauls Notes](https://tutorial.math.lamar.edu/classes/de/la_eigen.aspx#mm-1)/[Calculus II](https://tutorial.math.lamar.edu/classes/de/la_eigen.aspx#mm-18)/7\. Integration Techniques
- [7\.1 Integration by Parts](https://tutorial.math.lamar.edu/Classes/CalcII/IntegrationByParts.aspx)
- [7\.2 Integrals Involving Trig Functions](https://tutorial.math.lamar.edu/Classes/CalcII/IntegralsWithTrig.aspx)
- [7\.3 Trig Substitutions](https://tutorial.math.lamar.edu/Classes/CalcII/TrigSubstitutions.aspx)
- [7\.4 Partial Fractions](https://tutorial.math.lamar.edu/Classes/CalcII/PartialFractions.aspx)
- [7\.5 Integrals Involving Roots](https://tutorial.math.lamar.edu/Classes/CalcII/IntegralsWithRoots.aspx)
- [7\.6 Integrals Involving Quadratics](https://tutorial.math.lamar.edu/Classes/CalcII/IntegralsWithQuadratics.aspx)
- [7\.7 Integration Strategy](https://tutorial.math.lamar.edu/Classes/CalcII/IntegrationStrategy.aspx)
- [7\.8 Improper Integrals](https://tutorial.math.lamar.edu/Classes/CalcII/ImproperIntegrals.aspx)
- [7\.9 Comparison Test for Improper Integrals](https://tutorial.math.lamar.edu/Classes/CalcII/ImproperIntegralsCompTest.aspx)
- [7\.10 Approximating Definite Integrals](https://tutorial.math.lamar.edu/Classes/CalcII/ApproximatingDefIntegrals.aspx)
[Close submenu (8. Applications of Integrals)](https://tutorial.math.lamar.edu/classes/de/la_eigen.aspx#mm-18)[8\. Applications of Integrals](https://tutorial.math.lamar.edu/classes/de/la_eigen.aspx#mm-18)[Pauls Notes](https://tutorial.math.lamar.edu/classes/de/la_eigen.aspx#mm-1)/[Calculus II](https://tutorial.math.lamar.edu/classes/de/la_eigen.aspx#mm-18)/8\. Applications of Integrals
- [8\.1 Arc Length](https://tutorial.math.lamar.edu/Classes/CalcII/ArcLength.aspx)
- [8\.2 Surface Area](https://tutorial.math.lamar.edu/Classes/CalcII/SurfaceArea.aspx)
- [8\.3 Center of Mass](https://tutorial.math.lamar.edu/Classes/CalcII/CenterOfMass.aspx)
- [8\.4 Hydrostatic Pressure](https://tutorial.math.lamar.edu/Classes/CalcII/HydrostaticPressure.aspx)
- [8\.5 Probability](https://tutorial.math.lamar.edu/Classes/CalcII/Probability.aspx)
[Close submenu (9. Parametric Equations and Polar Coordinates)](https://tutorial.math.lamar.edu/classes/de/la_eigen.aspx#mm-18)[9\. Parametric Equations and Polar Coordinates](https://tutorial.math.lamar.edu/classes/de/la_eigen.aspx#mm-18)[Pauls Notes](https://tutorial.math.lamar.edu/classes/de/la_eigen.aspx#mm-1)/[Calculus II](https://tutorial.math.lamar.edu/classes/de/la_eigen.aspx#mm-18)/9\. Parametric Equations and Polar Coordinates
- [9\.1 Parametric Equations and Curves](https://tutorial.math.lamar.edu/Classes/CalcII/ParametricEqn.aspx)
- [9\.2 Tangents with Parametric Equations](https://tutorial.math.lamar.edu/Classes/CalcII/ParaTangent.aspx)
- [9\.3 Area with Parametric Equations](https://tutorial.math.lamar.edu/Classes/CalcII/ParaArea.aspx)
- [9\.4 Arc Length with Parametric Equations](https://tutorial.math.lamar.edu/Classes/CalcII/ParaArcLength.aspx)
- [9\.5 Surface Area with Parametric Equations](https://tutorial.math.lamar.edu/Classes/CalcII/ParaSurfaceArea.aspx)
- [9\.6 Polar Coordinates](https://tutorial.math.lamar.edu/Classes/CalcII/PolarCoordinates.aspx)
- [9\.7 Tangents with Polar Coordinates](https://tutorial.math.lamar.edu/Classes/CalcII/PolarTangents.aspx)
- [9\.8 Area with Polar Coordinates](https://tutorial.math.lamar.edu/Classes/CalcII/PolarArea.aspx)
- [9\.9 Arc Length with Polar Coordinates](https://tutorial.math.lamar.edu/Classes/CalcII/PolarArcLength.aspx)
- [9\.10 Surface Area with Polar Coordinates](https://tutorial.math.lamar.edu/Classes/CalcII/PolarSurfaceArea.aspx)
- [9\.11 Arc Length and Surface Area Revisited](https://tutorial.math.lamar.edu/Classes/CalcII/ArcLength_SurfaceArea.aspx)
[Close submenu (10. Series & Sequences)](https://tutorial.math.lamar.edu/classes/de/la_eigen.aspx#mm-18)[10\. Series & Sequences](https://tutorial.math.lamar.edu/classes/de/la_eigen.aspx#mm-18)[Pauls Notes](https://tutorial.math.lamar.edu/classes/de/la_eigen.aspx#mm-1)/[Calculus II](https://tutorial.math.lamar.edu/classes/de/la_eigen.aspx#mm-18)/10\. Series & Sequences
- [10\.1 Sequences](https://tutorial.math.lamar.edu/Classes/CalcII/Sequences.aspx)
- [10\.2 More on Sequences](https://tutorial.math.lamar.edu/Classes/CalcII/MoreSequences.aspx)
- [10\.3 Series - The Basics](https://tutorial.math.lamar.edu/Classes/CalcII/Series_Basics.aspx)
- [10\.4 Convergence/Divergence of Series](https://tutorial.math.lamar.edu/Classes/CalcII/ConvergenceOfSeries.aspx)
- [10\.5 Special Series](https://tutorial.math.lamar.edu/Classes/CalcII/Series_Special.aspx)
- [10\.6 Integral Test](https://tutorial.math.lamar.edu/Classes/CalcII/IntegralTest.aspx)
- [10\.7 Comparison Test/Limit Comparison Test](https://tutorial.math.lamar.edu/Classes/CalcII/SeriesCompTest.aspx)
- [10\.8 Alternating Series Test](https://tutorial.math.lamar.edu/Classes/CalcII/AlternatingSeries.aspx)
- [10\.9 Absolute Convergence](https://tutorial.math.lamar.edu/Classes/CalcII/AbsoluteConvergence.aspx)
- [10\.10 Ratio Test](https://tutorial.math.lamar.edu/Classes/CalcII/RatioTest.aspx)
- [10\.11 Root Test](https://tutorial.math.lamar.edu/Classes/CalcII/RootTest.aspx)
- [10\.12 Strategy for Series](https://tutorial.math.lamar.edu/Classes/CalcII/SeriesStrategy.aspx)
- [10\.13 Estimating the Value of a Series](https://tutorial.math.lamar.edu/Classes/CalcII/EstimatingSeries.aspx)
- [10\.14 Power Series](https://tutorial.math.lamar.edu/Classes/CalcII/PowerSeries.aspx)
- [10\.15 Power Series and Functions](https://tutorial.math.lamar.edu/Classes/CalcII/PowerSeriesandFunctions.aspx)
- [10\.16 Taylor Series](https://tutorial.math.lamar.edu/Classes/CalcII/TaylorSeries.aspx)
- [10\.17 Applications of Series](https://tutorial.math.lamar.edu/Classes/CalcII/TaylorSeriesApps.aspx)
- [10\.18 Binomial Series](https://tutorial.math.lamar.edu/Classes/CalcII/BinomialSeries.aspx)
[Close submenu (11. Vectors)](https://tutorial.math.lamar.edu/classes/de/la_eigen.aspx#mm-18)[11\. Vectors](https://tutorial.math.lamar.edu/classes/de/la_eigen.aspx#mm-18)[Pauls Notes](https://tutorial.math.lamar.edu/classes/de/la_eigen.aspx#mm-1)/[Calculus II](https://tutorial.math.lamar.edu/classes/de/la_eigen.aspx#mm-18)/11\. Vectors
- [11\.1 Vectors - The Basics](https://tutorial.math.lamar.edu/Classes/CalcII/Vectors_Basics.aspx)
- [11\.2 Vector Arithmetic](https://tutorial.math.lamar.edu/Classes/CalcII/VectorArithmetic.aspx)
- [11\.3 Dot Product](https://tutorial.math.lamar.edu/Classes/CalcII/DotProduct.aspx)
- [11\.4 Cross Product](https://tutorial.math.lamar.edu/Classes/CalcII/CrossProduct.aspx)
[Close submenu (12. 3-Dimensional Space)](https://tutorial.math.lamar.edu/classes/de/la_eigen.aspx#mm-18)[12\. 3-Dimensional Space](https://tutorial.math.lamar.edu/classes/de/la_eigen.aspx#mm-18)[Pauls Notes](https://tutorial.math.lamar.edu/classes/de/la_eigen.aspx#mm-1)/[Calculus II](https://tutorial.math.lamar.edu/classes/de/la_eigen.aspx#mm-18)/12\. 3-Dimensional Space
- [12\.1 The 3-D Coordinate System](https://tutorial.math.lamar.edu/Classes/CalcII/3DCoords.aspx)
- [12\.2 Equations of Lines](https://tutorial.math.lamar.edu/Classes/CalcII/EqnsOfLines.aspx)
- [12\.3 Equations of Planes](https://tutorial.math.lamar.edu/Classes/CalcII/EqnsOfPlanes.aspx)
- [12\.4 Quadric Surfaces](https://tutorial.math.lamar.edu/Classes/CalcII/QuadricSurfaces.aspx)
- [12\.5 Functions of Several Variables](https://tutorial.math.lamar.edu/Classes/CalcII/MultiVrbleFcns.aspx)
- [12\.6 Vector Functions](https://tutorial.math.lamar.edu/Classes/CalcII/VectorFunctions.aspx)
- [12\.7 Calculus with Vector Functions](https://tutorial.math.lamar.edu/Classes/CalcII/VectorFcnsCalculus.aspx)
- [12\.8 Tangent, Normal and Binormal Vectors](https://tutorial.math.lamar.edu/Classes/CalcII/TangentNormalVectors.aspx)
- [12\.9 Arc Length with Vector Functions](https://tutorial.math.lamar.edu/Classes/CalcII/VectorArcLength.aspx)
- [12\.10 Curvature](https://tutorial.math.lamar.edu/Classes/CalcII/Curvature.aspx)
- [12\.11 Velocity and Acceleration](https://tutorial.math.lamar.edu/Classes/CalcII/Velocity_Acceleration.aspx)
- [12\.12 Cylindrical Coordinates](https://tutorial.math.lamar.edu/Classes/CalcII/CylindricalCoords.aspx)
- [12\.13 Spherical Coordinates](https://tutorial.math.lamar.edu/Classes/CalcII/SphericalCoords.aspx)
[Close submenu (Calculus III)](https://tutorial.math.lamar.edu/classes/de/la_eigen.aspx#mm-1)[Calculus III](https://tutorial.math.lamar.edu/classes/de/la_eigen.aspx#mm-1)[Pauls Notes](https://tutorial.math.lamar.edu/classes/de/la_eigen.aspx#mm-1)/Calculus III
- [Open submenu (12. 3-Dimensional Space)](https://tutorial.math.lamar.edu/classes/de/la_eigen.aspx#mm-26)[12\. 3-Dimensional Space](https://tutorial.math.lamar.edu/Classes/CalcIII/3DSpace.aspx)
- [Open submenu (13. Partial Derivatives)](https://tutorial.math.lamar.edu/classes/de/la_eigen.aspx#mm-27)[13\. Partial Derivatives](https://tutorial.math.lamar.edu/Classes/CalcIII/PartialDerivsIntro.aspx)
- [Open submenu (14. Applications of Partial Derivatives)](https://tutorial.math.lamar.edu/classes/de/la_eigen.aspx#mm-28)[14\. Applications of Partial Derivatives](https://tutorial.math.lamar.edu/Classes/CalcIII/PartialDerivAppsIntro.aspx)
- [Open submenu (15. Multiple Integrals)](https://tutorial.math.lamar.edu/classes/de/la_eigen.aspx#mm-29)[15\. Multiple Integrals](https://tutorial.math.lamar.edu/Classes/CalcIII/MultipleIntegralsIntro.aspx)
- [Open submenu (16. Line Integrals)](https://tutorial.math.lamar.edu/classes/de/la_eigen.aspx#mm-30)[16\. Line Integrals](https://tutorial.math.lamar.edu/Classes/CalcIII/LineIntegralsIntro.aspx)
- [Open submenu (17.Surface Integrals)](https://tutorial.math.lamar.edu/classes/de/la_eigen.aspx#mm-31)[17\.Surface Integrals](https://tutorial.math.lamar.edu/Classes/CalcIII/SurfaceIntegralsIntro.aspx)
[Close submenu (12. 3-Dimensional Space)](https://tutorial.math.lamar.edu/classes/de/la_eigen.aspx#mm-25)[12\. 3-Dimensional Space](https://tutorial.math.lamar.edu/classes/de/la_eigen.aspx#mm-25)[Pauls Notes](https://tutorial.math.lamar.edu/classes/de/la_eigen.aspx#mm-1)/[Calculus III](https://tutorial.math.lamar.edu/classes/de/la_eigen.aspx#mm-25)/12\. 3-Dimensional Space
- [12\.1 The 3-D Coordinate System](https://tutorial.math.lamar.edu/Classes/CalcIII/3DCoords.aspx)
- [12\.2 Equations of Lines](https://tutorial.math.lamar.edu/Classes/CalcIII/EqnsOfLines.aspx)
- [12\.3 Equations of Planes](https://tutorial.math.lamar.edu/Classes/CalcIII/EqnsOfPlanes.aspx)
- [12\.4 Quadric Surfaces](https://tutorial.math.lamar.edu/Classes/CalcIII/QuadricSurfaces.aspx)
- [12\.5 Functions of Several Variables](https://tutorial.math.lamar.edu/Classes/CalcIII/MultiVrbleFcns.aspx)
- [12\.6 Vector Functions](https://tutorial.math.lamar.edu/Classes/CalcIII/VectorFunctions.aspx)
- [12\.7 Calculus with Vector Functions](https://tutorial.math.lamar.edu/Classes/CalcIII/VectorFcnsCalculus.aspx)
- [12\.8 Tangent, Normal and Binormal Vectors](https://tutorial.math.lamar.edu/Classes/CalcIII/TangentNormalVectors.aspx)
- [12\.9 Arc Length with Vector Functions](https://tutorial.math.lamar.edu/Classes/CalcIII/VectorArcLength.aspx)
- [12\.10 Curvature](https://tutorial.math.lamar.edu/Classes/CalcIII/Curvature.aspx)
- [12\.11 Velocity and Acceleration](https://tutorial.math.lamar.edu/Classes/CalcIII/Velocity_Acceleration.aspx)
- [12\.12 Cylindrical Coordinates](https://tutorial.math.lamar.edu/Classes/CalcIII/CylindricalCoords.aspx)
- [12\.13 Spherical Coordinates](https://tutorial.math.lamar.edu/Classes/CalcIII/SphericalCoords.aspx)
[Close submenu (13. Partial Derivatives)](https://tutorial.math.lamar.edu/classes/de/la_eigen.aspx#mm-25)[13\. Partial Derivatives](https://tutorial.math.lamar.edu/classes/de/la_eigen.aspx#mm-25)[Pauls Notes](https://tutorial.math.lamar.edu/classes/de/la_eigen.aspx#mm-1)/[Calculus III](https://tutorial.math.lamar.edu/classes/de/la_eigen.aspx#mm-25)/13\. Partial Derivatives
- [13\.1 Limits](https://tutorial.math.lamar.edu/Classes/CalcIII/Limits.aspx)
- [13\.2 Partial Derivatives](https://tutorial.math.lamar.edu/Classes/CalcIII/PartialDerivatives.aspx)
- [13\.3 Interpretations of Partial Derivatives](https://tutorial.math.lamar.edu/Classes/CalcIII/PartialDerivInterp.aspx)
- [13\.4 Higher Order Partial Derivatives](https://tutorial.math.lamar.edu/Classes/CalcIII/HighOrderPartialDerivs.aspx)
- [13\.5 Differentials](https://tutorial.math.lamar.edu/Classes/CalcIII/Differentials.aspx)
- [13\.6 Chain Rule](https://tutorial.math.lamar.edu/Classes/CalcIII/ChainRule.aspx)
- [13\.7 Directional Derivatives](https://tutorial.math.lamar.edu/Classes/CalcIII/DirectionalDeriv.aspx)
[Close submenu (14. Applications of Partial Derivatives)](https://tutorial.math.lamar.edu/classes/de/la_eigen.aspx#mm-25)[14\. Applications of Partial Derivatives](https://tutorial.math.lamar.edu/classes/de/la_eigen.aspx#mm-25)[Pauls Notes](https://tutorial.math.lamar.edu/classes/de/la_eigen.aspx#mm-1)/[Calculus III](https://tutorial.math.lamar.edu/classes/de/la_eigen.aspx#mm-25)/14\. Applications of Partial Derivatives
- [14\.1 Tangent Planes and Linear Approximations](https://tutorial.math.lamar.edu/Classes/CalcIII/TangentPlanes.aspx)
- [14\.2 Gradient Vector, Tangent Planes and Normal Lines](https://tutorial.math.lamar.edu/Classes/CalcIII/GradientVectorTangentPlane.aspx)
- [14\.3 Relative Minimums and Maximums](https://tutorial.math.lamar.edu/Classes/CalcIII/RelativeExtrema.aspx)
- [14\.4 Absolute Minimums and Maximums](https://tutorial.math.lamar.edu/Classes/CalcIII/AbsoluteExtrema.aspx)
- [14\.5 Lagrange Multipliers](https://tutorial.math.lamar.edu/Classes/CalcIII/LagrangeMultipliers.aspx)
[Close submenu (15. Multiple Integrals)](https://tutorial.math.lamar.edu/classes/de/la_eigen.aspx#mm-25)[15\. Multiple Integrals](https://tutorial.math.lamar.edu/classes/de/la_eigen.aspx#mm-25)[Pauls Notes](https://tutorial.math.lamar.edu/classes/de/la_eigen.aspx#mm-1)/[Calculus III](https://tutorial.math.lamar.edu/classes/de/la_eigen.aspx#mm-25)/15\. Multiple Integrals
- [15\.1 Double Integrals](https://tutorial.math.lamar.edu/Classes/CalcIII/DoubleIntegrals.aspx)
- [15\.2 Iterated Integrals](https://tutorial.math.lamar.edu/Classes/CalcIII/IteratedIntegrals.aspx)
- [15\.3 Double Integrals over General Regions](https://tutorial.math.lamar.edu/Classes/CalcIII/DIGeneralRegion.aspx)
- [15\.4 Double Integrals in Polar Coordinates](https://tutorial.math.lamar.edu/Classes/CalcIII/DIPolarCoords.aspx)
- [15\.5 Triple Integrals](https://tutorial.math.lamar.edu/Classes/CalcIII/TripleIntegrals.aspx)
- [15\.6 Triple Integrals in Cylindrical Coordinates](https://tutorial.math.lamar.edu/Classes/CalcIII/TICylindricalCoords.aspx)
- [15\.7 Triple Integrals in Spherical Coordinates](https://tutorial.math.lamar.edu/Classes/CalcIII/TISphericalCoords.aspx)
- [15\.8 Change of Variables](https://tutorial.math.lamar.edu/Classes/CalcIII/ChangeOfVariables.aspx)
- [15\.9 Surface Area](https://tutorial.math.lamar.edu/Classes/CalcIII/SurfaceArea.aspx)
- [15\.10 Area and Volume Revisited](https://tutorial.math.lamar.edu/Classes/CalcIII/Area_Volume.aspx)
[Close submenu (16. Line Integrals)](https://tutorial.math.lamar.edu/classes/de/la_eigen.aspx#mm-25)[16\. Line Integrals](https://tutorial.math.lamar.edu/classes/de/la_eigen.aspx#mm-25)[Pauls Notes](https://tutorial.math.lamar.edu/classes/de/la_eigen.aspx#mm-1)/[Calculus III](https://tutorial.math.lamar.edu/classes/de/la_eigen.aspx#mm-25)/16\. Line Integrals
- [16\.1 Vector Fields](https://tutorial.math.lamar.edu/Classes/CalcIII/VectorFields.aspx)
- [16\.2 Line Integrals - Part I](https://tutorial.math.lamar.edu/Classes/CalcIII/LineIntegralsPtI.aspx)
- [16\.3 Line Integrals - Part II](https://tutorial.math.lamar.edu/Classes/CalcIII/LineIntegralsPtII.aspx)
- [16\.4 Line Integrals of Vector Fields](https://tutorial.math.lamar.edu/Classes/CalcIII/LineIntegralsVectorFields.aspx)
- [16\.5 Fundamental Theorem for Line Integrals](https://tutorial.math.lamar.edu/Classes/CalcIII/FundThmLineIntegrals.aspx)
- [16\.6 Conservative Vector Fields](https://tutorial.math.lamar.edu/Classes/CalcIII/ConservativeVectorField.aspx)
- [16\.7 Green's Theorem](https://tutorial.math.lamar.edu/Classes/CalcIII/GreensTheorem.aspx)
[Close submenu (17.Surface Integrals)](https://tutorial.math.lamar.edu/classes/de/la_eigen.aspx#mm-25)[17\.Surface Integrals](https://tutorial.math.lamar.edu/classes/de/la_eigen.aspx#mm-25)[Pauls Notes](https://tutorial.math.lamar.edu/classes/de/la_eigen.aspx#mm-1)/[Calculus III](https://tutorial.math.lamar.edu/classes/de/la_eigen.aspx#mm-25)/17\.Surface Integrals
- [17\.1 Curl and Divergence](https://tutorial.math.lamar.edu/Classes/CalcIII/CurlDivergence.aspx)
- [17\.2 Parametric Surfaces](https://tutorial.math.lamar.edu/Classes/CalcIII/ParametricSurfaces.aspx)
- [17\.3 Surface Integrals](https://tutorial.math.lamar.edu/Classes/CalcIII/SurfaceIntegrals.aspx)
- [17\.4 Surface Integrals of Vector Fields](https://tutorial.math.lamar.edu/Classes/CalcIII/SurfIntVectorField.aspx)
- [17\.5 Stokes' Theorem](https://tutorial.math.lamar.edu/Classes/CalcIII/StokesTheorem.aspx)
- [17\.6 Divergence Theorem](https://tutorial.math.lamar.edu/Classes/CalcIII/DivergenceTheorem.aspx)
[Close submenu (Differential Equations)](https://tutorial.math.lamar.edu/classes/de/la_eigen.aspx#mm-1)[Differential Equations](https://tutorial.math.lamar.edu/classes/de/la_eigen.aspx#mm-1)[Pauls Notes](https://tutorial.math.lamar.edu/classes/de/la_eigen.aspx#mm-1)/Differential Equations
- [Open submenu (1. Basic Concepts)](https://tutorial.math.lamar.edu/classes/de/la_eigen.aspx#mm-33)[1\. Basic Concepts](https://tutorial.math.lamar.edu/Classes/DE/IntroBasic.aspx)
- [Open submenu (2. First Order DE's)](https://tutorial.math.lamar.edu/classes/de/la_eigen.aspx#mm-34)[2\. First Order DE's](https://tutorial.math.lamar.edu/Classes/DE/IntroFirstOrder.aspx)
- [Open submenu (3. Second Order DE's)](https://tutorial.math.lamar.edu/classes/de/la_eigen.aspx#mm-35)[3\. Second Order DE's](https://tutorial.math.lamar.edu/Classes/DE/IntroSecondOrder.aspx)
- [Open submenu (4. Laplace Transforms)](https://tutorial.math.lamar.edu/classes/de/la_eigen.aspx#mm-36)[4\. Laplace Transforms](https://tutorial.math.lamar.edu/Classes/DE/LaplaceIntro.aspx)
- [Open submenu (5. Systems of DE's)](https://tutorial.math.lamar.edu/classes/de/la_eigen.aspx#mm-37)[5\. Systems of DE's](https://tutorial.math.lamar.edu/Classes/DE/SystemsIntro.aspx)
- [Open submenu (6. Series Solutions to DE's)](https://tutorial.math.lamar.edu/classes/de/la_eigen.aspx#mm-38)[6\. Series Solutions to DE's](https://tutorial.math.lamar.edu/Classes/DE/SeriesIntro.aspx)
- [Open submenu (7. Higher Order Differential Equations)](https://tutorial.math.lamar.edu/classes/de/la_eigen.aspx#mm-39)[7\. Higher Order Differential Equations](https://tutorial.math.lamar.edu/Classes/DE/IntroHigherOrder.aspx)
- [Open submenu (8. Boundary Value Problems & Fourier Series)](https://tutorial.math.lamar.edu/classes/de/la_eigen.aspx#mm-40)[8\. Boundary Value Problems & Fourier Series](https://tutorial.math.lamar.edu/Classes/DE/IntroBVP.aspx)
- [Open submenu (9. Partial Differential Equations )](https://tutorial.math.lamar.edu/classes/de/la_eigen.aspx#mm-41)[9\. Partial Differential Equations](https://tutorial.math.lamar.edu/Classes/DE/IntroPDE.aspx)
[Close submenu (1. Basic Concepts)](https://tutorial.math.lamar.edu/classes/de/la_eigen.aspx#mm-32)[1\. Basic Concepts](https://tutorial.math.lamar.edu/classes/de/la_eigen.aspx#mm-32)[Pauls Notes](https://tutorial.math.lamar.edu/classes/de/la_eigen.aspx#mm-1)/[Differential Equations](https://tutorial.math.lamar.edu/classes/de/la_eigen.aspx#mm-32)/1\. Basic Concepts
- [1\.1 Definitions](https://tutorial.math.lamar.edu/Classes/DE/Definitions.aspx)
- [1\.2 Direction Fields](https://tutorial.math.lamar.edu/Classes/DE/DirectionFields.aspx)
- [1\.3 Final Thoughts](https://tutorial.math.lamar.edu/Classes/DE/FinalThoughts.aspx)
[Close submenu (2. First Order DE's)](https://tutorial.math.lamar.edu/classes/de/la_eigen.aspx#mm-32)[2\. First Order DE's](https://tutorial.math.lamar.edu/classes/de/la_eigen.aspx#mm-32)[Pauls Notes](https://tutorial.math.lamar.edu/classes/de/la_eigen.aspx#mm-1)/[Differential Equations](https://tutorial.math.lamar.edu/classes/de/la_eigen.aspx#mm-32)/2\. First Order DE's
- [2\.1 Linear Equations](https://tutorial.math.lamar.edu/Classes/DE/Linear.aspx)
- [2\.2 Separable Equations](https://tutorial.math.lamar.edu/Classes/DE/Separable.aspx)
- [2\.3 Exact Equations](https://tutorial.math.lamar.edu/Classes/DE/Exact.aspx)
- [2\.4 Bernoulli Differential Equations](https://tutorial.math.lamar.edu/Classes/DE/Bernoulli.aspx)
- [2\.5 Substitutions](https://tutorial.math.lamar.edu/Classes/DE/Substitutions.aspx)
- [2\.6 Intervals of Validity](https://tutorial.math.lamar.edu/Classes/DE/IoV.aspx)
- [2\.7 Modeling with First Order DE's](https://tutorial.math.lamar.edu/Classes/DE/Modeling.aspx)
- [2\.8 Equilibrium Solutions](https://tutorial.math.lamar.edu/Classes/DE/EquilibriumSolutions.aspx)
- [2\.9 Euler's Method](https://tutorial.math.lamar.edu/Classes/DE/EulersMethod.aspx)
[Close submenu (3. Second Order DE's)](https://tutorial.math.lamar.edu/classes/de/la_eigen.aspx#mm-32)[3\. Second Order DE's](https://tutorial.math.lamar.edu/classes/de/la_eigen.aspx#mm-32)[Pauls Notes](https://tutorial.math.lamar.edu/classes/de/la_eigen.aspx#mm-1)/[Differential Equations](https://tutorial.math.lamar.edu/classes/de/la_eigen.aspx#mm-32)/3\. Second Order DE's
- [3\.1 Basic Concepts](https://tutorial.math.lamar.edu/Classes/DE/SecondOrderConcepts.aspx)
- [3\.2 Real & Distinct Roots](https://tutorial.math.lamar.edu/Classes/DE/RealRoots.aspx)
- [3\.3 Complex Roots](https://tutorial.math.lamar.edu/Classes/DE/ComplexRoots.aspx)
- [3\.4 Repeated Roots](https://tutorial.math.lamar.edu/Classes/DE/RepeatedRoots.aspx)
- [3\.5 Reduction of Order](https://tutorial.math.lamar.edu/Classes/DE/ReductionofOrder.aspx)
- [3\.6 Fundamental Sets of Solutions](https://tutorial.math.lamar.edu/Classes/DE/FundamentalSetsofSolutions.aspx)
- [3\.7 More on the Wronskian](https://tutorial.math.lamar.edu/Classes/DE/Wronskian.aspx)
- [3\.8 Nonhomogeneous Differential Equations](https://tutorial.math.lamar.edu/Classes/DE/NonhomogeneousDE.aspx)
- [3\.9 Undetermined Coefficients](https://tutorial.math.lamar.edu/Classes/DE/UndeterminedCoefficients.aspx)
- [3\.10 Variation of Parameters](https://tutorial.math.lamar.edu/Classes/DE/VariationofParameters.aspx)
- [3\.11 Mechanical Vibrations](https://tutorial.math.lamar.edu/Classes/DE/Vibrations.aspx)
[Close submenu (4. Laplace Transforms)](https://tutorial.math.lamar.edu/classes/de/la_eigen.aspx#mm-32)[4\. Laplace Transforms](https://tutorial.math.lamar.edu/classes/de/la_eigen.aspx#mm-32)[Pauls Notes](https://tutorial.math.lamar.edu/classes/de/la_eigen.aspx#mm-1)/[Differential Equations](https://tutorial.math.lamar.edu/classes/de/la_eigen.aspx#mm-32)/4\. Laplace Transforms
- [4\.1 The Definition](https://tutorial.math.lamar.edu/Classes/DE/LaplaceDefinition.aspx)
- [4\.2 Laplace Transforms](https://tutorial.math.lamar.edu/Classes/DE/LaplaceTransforms.aspx)
- [4\.3 Inverse Laplace Transforms](https://tutorial.math.lamar.edu/Classes/DE/InverseTransforms.aspx)
- [4\.4 Step Functions](https://tutorial.math.lamar.edu/Classes/DE/StepFunctions.aspx)
- [4\.5 Solving IVP's with Laplace Transforms](https://tutorial.math.lamar.edu/Classes/DE/IVPWithLaplace.aspx)
- [4\.6 Nonconstant Coefficient IVP's](https://tutorial.math.lamar.edu/Classes/DE/IVPWithNonConstantCoefficient.aspx)
- [4\.7 IVP's With Step Functions](https://tutorial.math.lamar.edu/Classes/DE/IVPWithStepFunction.aspx)
- [4\.8 Dirac Delta Function](https://tutorial.math.lamar.edu/Classes/DE/DiracDeltaFunction.aspx)
- [4\.9 Convolution Integrals](https://tutorial.math.lamar.edu/Classes/DE/ConvolutionIntegrals.aspx)
- [4\.10 Table Of Laplace Transforms](https://tutorial.math.lamar.edu/Classes/DE/Laplace_Table.aspx)
[Close submenu (5. Systems of DE's)](https://tutorial.math.lamar.edu/classes/de/la_eigen.aspx#mm-32)[5\. Systems of DE's](https://tutorial.math.lamar.edu/classes/de/la_eigen.aspx#mm-32)[Pauls Notes](https://tutorial.math.lamar.edu/classes/de/la_eigen.aspx#mm-1)/[Differential Equations](https://tutorial.math.lamar.edu/classes/de/la_eigen.aspx#mm-32)/5\. Systems of DE's
- [5\.1 Review : Systems of Equations](https://tutorial.math.lamar.edu/Classes/DE/LA_Systems.aspx)
- [5\.2 Review : Matrices & Vectors](https://tutorial.math.lamar.edu/Classes/DE/LA_Matrix.aspx)
- [5\.3 Review : Eigenvalues & Eigenvectors](https://tutorial.math.lamar.edu/Classes/DE/LA_Eigen.aspx)
- [5\.4 Systems of Differential Equations](https://tutorial.math.lamar.edu/Classes/DE/SystemsDE.aspx)
- [5\.5 Solutions to Systems](https://tutorial.math.lamar.edu/Classes/DE/SolutionsToSystems.aspx)
- [5\.6 Phase Plane](https://tutorial.math.lamar.edu/Classes/DE/PhasePlane.aspx)
- [5\.7 Real Eigenvalues](https://tutorial.math.lamar.edu/Classes/DE/RealEigenvalues.aspx)
- [5\.8 Complex Eigenvalues](https://tutorial.math.lamar.edu/Classes/DE/ComplexEigenvalues.aspx)
- [5\.9 Repeated Eigenvalues](https://tutorial.math.lamar.edu/Classes/DE/RepeatedEigenvalues.aspx)
- [5\.10 Nonhomogeneous Systems](https://tutorial.math.lamar.edu/Classes/DE/NonhomogeneousSystems.aspx)
- [5\.11 Laplace Transforms](https://tutorial.math.lamar.edu/Classes/DE/SystemsLaplace.aspx)
- [5\.12 Modeling](https://tutorial.math.lamar.edu/Classes/DE/SystemsModeling.aspx)
[Close submenu (6. Series Solutions to DE's)](https://tutorial.math.lamar.edu/classes/de/la_eigen.aspx#mm-32)[6\. Series Solutions to DE's](https://tutorial.math.lamar.edu/classes/de/la_eigen.aspx#mm-32)[Pauls Notes](https://tutorial.math.lamar.edu/classes/de/la_eigen.aspx#mm-1)/[Differential Equations](https://tutorial.math.lamar.edu/classes/de/la_eigen.aspx#mm-32)/6\. Series Solutions to DE's
- [6\.1 Review : Power Series](https://tutorial.math.lamar.edu/Classes/DE/PowerSeries.aspx)
- [6\.2 Review : Taylor Series](https://tutorial.math.lamar.edu/Classes/DE/TaylorSeries.aspx)
- [6\.3 Series Solutions](https://tutorial.math.lamar.edu/Classes/DE/SeriesSolutions.aspx)
- [6\.4 Euler Equations](https://tutorial.math.lamar.edu/Classes/DE/EulerEquations.aspx)
[Close submenu (7. Higher Order Differential Equations)](https://tutorial.math.lamar.edu/classes/de/la_eigen.aspx#mm-32)[7\. Higher Order Differential Equations](https://tutorial.math.lamar.edu/classes/de/la_eigen.aspx#mm-32)[Pauls Notes](https://tutorial.math.lamar.edu/classes/de/la_eigen.aspx#mm-1)/[Differential Equations](https://tutorial.math.lamar.edu/classes/de/la_eigen.aspx#mm-32)/7\. Higher Order Differential Equations
- [7\.1 Basic Concepts for *n*th Order Linear Equations](https://tutorial.math.lamar.edu/Classes/DE/HOBasicConcepts.aspx)
- [7\.2 Linear Homogeneous Differential Equations](https://tutorial.math.lamar.edu/Classes/DE/HOHomogeneousDE.aspx)
- [7\.3 Undetermined Coefficients](https://tutorial.math.lamar.edu/Classes/DE/HOUndeterminedCoeff.aspx)
- [7\.4 Variation of Parameters](https://tutorial.math.lamar.edu/Classes/DE/HOVariationOfParam.aspx)
- [7\.5 Laplace Transforms](https://tutorial.math.lamar.edu/Classes/DE/HOLaplaceTransforms.aspx)
- [7\.6 Systems of Differential Equations](https://tutorial.math.lamar.edu/Classes/DE/HOSystems.aspx)
- [7\.7 Series Solutions](https://tutorial.math.lamar.edu/Classes/DE/HOSeries.aspx)
[Close submenu (8. Boundary Value Problems & Fourier Series)](https://tutorial.math.lamar.edu/classes/de/la_eigen.aspx#mm-32)[8\. Boundary Value Problems & Fourier Series](https://tutorial.math.lamar.edu/classes/de/la_eigen.aspx#mm-32)[Pauls Notes](https://tutorial.math.lamar.edu/classes/de/la_eigen.aspx#mm-1)/[Differential Equations](https://tutorial.math.lamar.edu/classes/de/la_eigen.aspx#mm-32)/8\. Boundary Value Problems & Fourier Series
- [8\.1 Boundary Value Problems](https://tutorial.math.lamar.edu/Classes/DE/BoundaryValueProblem.aspx)
- [8\.2 Eigenvalues and Eigenfunctions](https://tutorial.math.lamar.edu/Classes/DE/BVPEvals.aspx)
- [8\.3 Periodic Functions & Orthogonal Functions](https://tutorial.math.lamar.edu/Classes/DE/PeriodicOrthogonal.aspx)
- [8\.4 Fourier Sine Series](https://tutorial.math.lamar.edu/Classes/DE/FourierSineSeries.aspx)
- [8\.5 Fourier Cosine Series](https://tutorial.math.lamar.edu/Classes/DE/FourierCosineSeries.aspx)
- [8\.6 Fourier Series](https://tutorial.math.lamar.edu/Classes/DE/FourierSeries.aspx)
- [8\.7 Convergence of Fourier Series](https://tutorial.math.lamar.edu/Classes/DE/ConvergenceFourierSeries.aspx)
[Close submenu (9. Partial Differential Equations )](https://tutorial.math.lamar.edu/classes/de/la_eigen.aspx#mm-32)[9\. Partial Differential Equations](https://tutorial.math.lamar.edu/classes/de/la_eigen.aspx#mm-32) [Pauls Notes](https://tutorial.math.lamar.edu/classes/de/la_eigen.aspx#mm-1)/[Differential Equations](https://tutorial.math.lamar.edu/classes/de/la_eigen.aspx#mm-32)/9\. Partial Differential Equations
- [9\.1 The Heat Equation](https://tutorial.math.lamar.edu/Classes/DE/TheHeatEquation.aspx)
- [9\.2 The Wave Equation](https://tutorial.math.lamar.edu/Classes/DE/TheWaveEquation.aspx)
- [9\.3 Terminology](https://tutorial.math.lamar.edu/Classes/DE/PDETerminology.aspx)
- [9\.4 Separation of Variables](https://tutorial.math.lamar.edu/Classes/DE/SeparationofVariables.aspx)
- [9\.5 Solving the Heat Equation](https://tutorial.math.lamar.edu/Classes/DE/SolvingHeatEquation.aspx)
- [9\.6 Heat Equation with Non-Zero Temperature Boundaries](https://tutorial.math.lamar.edu/Classes/DE/HeatEqnNonZero.aspx)
- [9\.7 Laplace's Equation](https://tutorial.math.lamar.edu/Classes/DE/LaplacesEqn.aspx)
- [9\.8 Vibrating String](https://tutorial.math.lamar.edu/Classes/DE/VibratingString.aspx)
- [9\.9 Summary of Separation of Variables](https://tutorial.math.lamar.edu/Classes/DE/PDESummary.aspx)
[Close submenu (Algebra & Trig Review)](https://tutorial.math.lamar.edu/classes/de/la_eigen.aspx#mm-1)[Algebra & Trig Review](https://tutorial.math.lamar.edu/classes/de/la_eigen.aspx#mm-1)[Pauls Notes](https://tutorial.math.lamar.edu/classes/de/la_eigen.aspx#mm-1)/Algebra & Trig Review
- [Open submenu (1. Algebra)](https://tutorial.math.lamar.edu/classes/de/la_eigen.aspx#mm-43)[1\. Algebra](https://tutorial.math.lamar.edu/Extras/AlgebraTrigReview/AlgebraIntro.aspx)
- [Open submenu (2. Trigonometry)](https://tutorial.math.lamar.edu/classes/de/la_eigen.aspx#mm-44)[2\. Trigonometry](https://tutorial.math.lamar.edu/Extras/AlgebraTrigReview/TrigIntro.aspx)
- [Open submenu (3. Exponentials & Logarithms)](https://tutorial.math.lamar.edu/classes/de/la_eigen.aspx#mm-45)[3\. Exponentials & Logarithms](https://tutorial.math.lamar.edu/Extras/AlgebraTrigReview/ExpLogIntro.aspx)
[Close submenu (1. Algebra)](https://tutorial.math.lamar.edu/classes/de/la_eigen.aspx#mm-42)[1\. Algebra](https://tutorial.math.lamar.edu/classes/de/la_eigen.aspx#mm-42)[Pauls Notes](https://tutorial.math.lamar.edu/classes/de/la_eigen.aspx#mm-1)/[Algebra & Trig Review](https://tutorial.math.lamar.edu/classes/de/la_eigen.aspx#mm-42)/1\. Algebra
- [1\.1 Exponents](https://tutorial.math.lamar.edu/Extras/AlgebraTrigReview/Exponents.aspx)
- [1\.2 Absolute Value](https://tutorial.math.lamar.edu/Extras/AlgebraTrigReview/AbsoluteValue.aspx)
- [1\.3 Radicals](https://tutorial.math.lamar.edu/Extras/AlgebraTrigReview/Radicals.aspx)
- [1\.4 Rationalizing](https://tutorial.math.lamar.edu/Extras/AlgebraTrigReview/Rationalizing.aspx)
- [1\.5 Functions](https://tutorial.math.lamar.edu/Extras/AlgebraTrigReview/Functions.aspx)
- [1\.6 Multiplying Polynomials](https://tutorial.math.lamar.edu/Extras/AlgebraTrigReview/MultPoly.aspx)
- [1\.7 Factoring](https://tutorial.math.lamar.edu/Extras/AlgebraTrigReview/Factoring.aspx)
- [1\.8 Simplifying Rational Expressions](https://tutorial.math.lamar.edu/Extras/AlgebraTrigReview/SimpRatExp.aspx)
- [1\.9 Graphing and Common Graphs](https://tutorial.math.lamar.edu/Extras/AlgebraTrigReview/Graphing.aspx)
- [1\.10 Solving Equations, Part I](https://tutorial.math.lamar.edu/Extras/AlgebraTrigReview/SolveEqnPtI.aspx)
- [1\.11 Solving Equations, Part II](https://tutorial.math.lamar.edu/Extras/AlgebraTrigReview/SolveEqnPtII.aspx)
- [1\.12 Solving Systems of Equations](https://tutorial.math.lamar.edu/Extras/AlgebraTrigReview/SolveSystems.aspx)
- [1\.13 Solving Inequalities](https://tutorial.math.lamar.edu/Extras/AlgebraTrigReview/SolveIneq.aspx)
- [1\.14 Absolute Value Equations and Inequalities](https://tutorial.math.lamar.edu/Extras/AlgebraTrigReview/SolveAbsValue.aspx)
[Close submenu (2. Trigonometry)](https://tutorial.math.lamar.edu/classes/de/la_eigen.aspx#mm-42)[2\. Trigonometry](https://tutorial.math.lamar.edu/classes/de/la_eigen.aspx#mm-42)[Pauls Notes](https://tutorial.math.lamar.edu/classes/de/la_eigen.aspx#mm-1)/[Algebra & Trig Review](https://tutorial.math.lamar.edu/classes/de/la_eigen.aspx#mm-42)/2\. Trigonometry
- [2\.1 Trig Function Evaluation](https://tutorial.math.lamar.edu/Extras/AlgebraTrigReview/TrigFunctions.aspx)
- [2\.2 Graphs of Trig Functions](https://tutorial.math.lamar.edu/Extras/AlgebraTrigReview/TrigGraphs.aspx)
- [2\.3 Trig Formulas](https://tutorial.math.lamar.edu/Extras/AlgebraTrigReview/TrigFormulas.aspx)
- [2\.4 Solving Trig Equations](https://tutorial.math.lamar.edu/Extras/AlgebraTrigReview/SolveTrigEqn.aspx)
- [2\.5 Inverse Trig Functions](https://tutorial.math.lamar.edu/Extras/AlgebraTrigReview/InverseTrig.aspx)
[Close submenu (3. Exponentials & Logarithms)](https://tutorial.math.lamar.edu/classes/de/la_eigen.aspx#mm-42)[3\. Exponentials & Logarithms](https://tutorial.math.lamar.edu/classes/de/la_eigen.aspx#mm-42)[Pauls Notes](https://tutorial.math.lamar.edu/classes/de/la_eigen.aspx#mm-1)/[Algebra & Trig Review](https://tutorial.math.lamar.edu/classes/de/la_eigen.aspx#mm-42)/3\. Exponentials & Logarithms
- [3\.1 Basic Exponential Functions](https://tutorial.math.lamar.edu/Extras/AlgebraTrigReview/ExponentialFcns.aspx)
- [3\.2 Basic Logarithm Functions](https://tutorial.math.lamar.edu/Extras/AlgebraTrigReview/LogarithmFcns.aspx)
- [3\.3 Logarithm Properties](https://tutorial.math.lamar.edu/Extras/AlgebraTrigReview/LogProperties.aspx)
- [3\.4 Simplifying Logarithms](https://tutorial.math.lamar.edu/Extras/AlgebraTrigReview/SimpLogs.aspx)
- [3\.5 Solving Exponential Equations](https://tutorial.math.lamar.edu/Extras/AlgebraTrigReview/SolveExpEqn.aspx)
- [3\.6 Solving Logarithm Equations](https://tutorial.math.lamar.edu/Extras/AlgebraTrigReview/SolveLogEqn.aspx)
[Close submenu (Common Math Errors)](https://tutorial.math.lamar.edu/classes/de/la_eigen.aspx#mm-1)[Common Math Errors](https://tutorial.math.lamar.edu/classes/de/la_eigen.aspx#mm-1)[Pauls Notes](https://tutorial.math.lamar.edu/classes/de/la_eigen.aspx#mm-1)/Common Math Errors
- [1\. General Errors](https://tutorial.math.lamar.edu/Extras/CommonErrors/GeneralErrors.aspx)
- [2\. Algebra Errors](https://tutorial.math.lamar.edu/Extras/CommonErrors/AlgebraErrors.aspx)
- [3\. Trig Errors](https://tutorial.math.lamar.edu/Extras/CommonErrors/TrigErrors.aspx)
- [4\. Common Errors](https://tutorial.math.lamar.edu/Extras/CommonErrors/CommonErrors.aspx)
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[Close submenu (Complex Number Primer)](https://tutorial.math.lamar.edu/classes/de/la_eigen.aspx#mm-1)[Complex Number Primer](https://tutorial.math.lamar.edu/classes/de/la_eigen.aspx#mm-1)[Pauls Notes](https://tutorial.math.lamar.edu/classes/de/la_eigen.aspx#mm-1)/Complex Number Primer
- [1\. The Definition](https://tutorial.math.lamar.edu/Extras/ComplexPrimer/Definition.aspx)
- [2\. Arithmetic](https://tutorial.math.lamar.edu/Extras/ComplexPrimer/Arithmetic.aspx)
- [3\. Conjugate and Modulus](https://tutorial.math.lamar.edu/Extras/ComplexPrimer/ConjugateModulus.aspx)
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[Close submenu (How To Study Math)](https://tutorial.math.lamar.edu/classes/de/la_eigen.aspx#mm-1)[How To Study Math](https://tutorial.math.lamar.edu/classes/de/la_eigen.aspx#mm-1)[Pauls Notes](https://tutorial.math.lamar.edu/classes/de/la_eigen.aspx#mm-1)/How To Study Math
- [1\. General Tips](https://tutorial.math.lamar.edu/Extras/StudyMath/GeneralTips.aspx)
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- [8\. Learn From Your Errors](https://tutorial.math.lamar.edu/Extras/StudyMath/Errors.aspx)
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### Section 5.3 : Review : Eigenvalues & Eigenvectors
If you get nothing out of this quick review of linear algebra you must get this section. Without this section you will not be able to do any of the differential equations work that is in this chapter.
So, let’s start with the following. If we multiply an n×n n × n matrix by an n×1 n × 1 vector we will get a new n×1 n × 1 vector back. In other words,
A→η\=→y
A
η
→
\=
y
→
What we want to know is if it is possible for the following to happen. Instead of just getting a brand new vector out of the multiplication is it possible instead to get the following,
A→η\=λ→η(1)
(1)
A
η
→
\=
λ
η
→
In other words, is it possible, at least for certain λ λ and →η η →, to have matrix multiplication be the same as just multiplying the vector by a constant? Of course, we probably wouldn’t be talking about this if the answer was no. So, it is possible for this to happen, however, it won’t happen for just any value of λ λ or →η η →. If we do happen to have a λ λ and →η η → for which this works (and they will always come in pairs) then we call λ λ an **eigenvalue** of A A and →η η → an **eigenvector** of A A.
So, how do we go about finding the eigenvalues and eigenvectors for a matrix? Well first notice that if →η\=→0 η → \= 0 → then [(1)](https://tutorial.math.lamar.edu/classes/de/la_eigen.aspx#mjx-eqn-eqeq1) (1) is going to be true for any value of λ λ and so we are going to make the assumption that →η≠→0 η → ≠ 0 →. With that out of the way let’s rewrite [(1)](https://tutorial.math.lamar.edu/classes/de/la_eigen.aspx#mjx-eqn-eqeq1) (1) a little.
A→η−λ→η\=→0A→η−λIn→η\=→0(A−λIn)→η\=→0
A
η
→
−
λ
η
→
\=
0
→
A
η
→
−
λ
I
n
η
→
\=
0
→
(
A
−
λ
I
n
)
η
→
\=
0
→
Notice that before we factored out the →η η → we added in the appropriately sized identity matrix. This is equivalent to multiplying things by a one and so doesn’t change the value of anything. We needed to do this because without it we would have had the difference of a matrix, A A, and a constant, λ λ, and this can’t be done. We now have the difference of two matrices of the same size which can be done.
So, with this rewrite we see that
(A−λIn)→η\=→0(2)
(2)
(
A
−
λ
I
n
)
η
→
\=
0
→
is equivalent to [(1)](https://tutorial.math.lamar.edu/classes/de/la_eigen.aspx#mjx-eqn-eqeq1) (1). In order to find the eigenvectors for a matrix we will need to solve a homogeneous system. Recall the [fact](https://tutorial.math.lamar.edu/classes/de/LA_Matrix.aspx#System_Fact_2) from the previous section that we know that we will either have exactly one solution (→η\=→0 η → \= 0 →) or we will have infinitely many nonzero solutions. Since we’ve already said that we don’t want →η\=→0 η → \= 0 → this means that we want the second case.
Knowing this will allow us to find the eigenvalues for a matrix. Recall from this fact that we will get the second case only if the matrix in the system is singular. Therefore, we will need to determine the values of λ λ for which we get,
det(A−λI)\=0
det
(
A
−
λ
I
)
\=
0
Once we have the eigenvalues we can then go back and determine the eigenvectors for each eigenvalue. Let’s take a look at a couple of quick facts about eigenvalues and eigenvectors.
#### Fact
If A A is an n×n n × n matrix then det(A−λI)\=0 det ( A − λ I ) \= 0 is an nth n th degree polynomial. This polynomial is called the **characteristic polynomial**.
To find eigenvalues of a matrix all we need to do is solve a polynomial. That’s generally not too bad provided we keep n n small. Likewise this fact also tells us that for an n×n n × n matrix, A A, we will have n n eigenvalues if we include all repeated eigenvalues.
#### Fact
If λ1,λ2,…,λn λ 1 , λ 2 , … , λ n is the complete list of eigenvalues for A A (including all repeated eigenvalues) then,
1. If
λ
λ
occurs only once in the list then we call
λ
λ
**simple**.
2. If
λ
λ
occurs
k\>1
k
\>
1
times in the list then we say that
λ
λ
has **multiplicity k k**.
3. If
λ1,λ2,…,λk
λ
1
,
λ
2
,
…
,
λ
k
(
k≤n
k
≤
n
) are the simple eigenvalues in the list with corresponding eigenvectors
→η(1)
η
→
(
1
)
,
→η(2)
η
→
(
2
)
, …,
→η(k)
η
→
(
k
)
then the eigenvectors are all linearly independent.
4. If
λ
λ
is an eigenvalue of multiplicity
k\>1
k
\>
1
then
λ
λ
will have anywhere from 1 to
k
k
linearly independent eigenvectors.
The usefulness of these facts will become apparent when we get back into differential equations since in that work we will want linearly independent solutions.
Let’s work a couple of examples now to see how we actually go about finding eigenvalues and eigenvectors.
Example 1 Find the eigenvalues and eigenvectors of the following matrix.
A\=(27−1−6)
A
\=
(
2
7
−
1
−
6
)
Show Solution
The first thing that we need to do is find the eigenvalues. That means we need the following matrix,
A−λI\=(27−1−6)−λ(1001)\=(2−λ7−1−6−λ)
A
−
λ
I
\=
(
2
7
−
1
−
6
)
−
λ
(
1
0
0
1
)
\=
(
2
−
λ
7
−
1
−
6
−
λ
)
In particular we need to determine where the determinant of this matrix is zero.
det(A−λI)\=(2−λ)(−6−λ)\+7\=λ2\+4λ−5\=(λ\+5)(λ−1)
det
(
A
−
λ
I
)
\=
(
2
−
λ
)
(
−
6
−
λ
)
\+
7
\=
λ
2
\+
4
λ
−
5
\=
(
λ
\+
5
)
(
λ
−
1
)
So, it looks like we will have two simple eigenvalues for this matrix, λ1\=−5 λ 1 \= − 5 and λ2\=1 λ 2 \= 1. We will now need to find the eigenvectors for each of these. Also note that according to the fact above, the two eigenvectors should be linearly independent.
To find the eigenvectors we simply plug in each eigenvalue into and solve. So, let’s do that.
λ1\=−5 λ 1 \= − 5 :
In this case we need to solve the following system.
(77−1−1)→η\=(00)
(
7
7
−
1
−
1
)
η
→
\=
(
0
0
)
Recall that officially to solve this system we use the following augmented matrix.
(770−1−10)17R1\+R2⇒(770000)
(
7
7
0
−
1
−
1
0
)
1
7
R
1
\+
R
2
⇒
(
7
7
0
0
0
0
)
Upon reducing down we see that we get a single equation
7η1\+7η2\=0⇒η1\=−η2
7
η
1
\+
7
η
2
\=
0
⇒
η
1
\=
−
η
2
that will yield an infinite number of solutions. This is expected behavior. Recall that we picked the eigenvalues so that the matrix would be singular and so we would get infinitely many solutions.
Notice as well that we could have identified this from the original system. This won’t always be the case, but in the 2×2 2 × 2 case we can see from the system that one row will be a multiple of the other and so we will get infinite solutions. From this point on we won’t be actually solving systems in these cases. We will just go straight to the equation and we can use either of the two rows for this equation.
Now, let’s get back to the eigenvector, since that is what we were after. In general then the eigenvector will be any vector that satisfies the following,
→η\=(η1η2)\=(−η2η2),η2≠0
η
→
\=
(
η
1
η
2
)
\=
(
−
η
2
η
2
)
,
η
2
≠
0
To get this we used the solution to the equation that we found above.
We really don’t want a general eigenvector however so we will pick a value for η2 η 2 to get a specific eigenvector. We can choose anything (except η2\=0 η 2 \= 0), so pick something that will make the eigenvector “nice”. Note as well that since we’ve already assumed that the eigenvector is not zero we must choose a value that will not give us zero, which is why we want to avoid η2\=0 η 2 \= 0 in this case. Here’s the eigenvector for this eigenvalue.
→η(1)\=(−11),using η2\=1
η
→
(
1
)
\=
(
−
1
1
)
,
using
η
2
\=
1
Now we get to do this all over again for the second eigenvalue.
λ2\=1 λ 2 \= 1 :
We’ll do much less work with this part than we did with the previous part. We will need to solve the following system.
(17−1−7)→η\=(00)
(
1
7
−
1
−
7
)
η
→
\=
(
0
0
)
Clearly both rows are multiples of each other and so we will get infinitely many solutions. We can choose to work with either row. We’ll run with the first because to avoid having too many minus signs floating around. Doing this gives us,
η1\+7η2\=0η1\=−7η2
η
1
\+
7
η
2
\=
0
η
1
\=
−
7
η
2
Note that we can solve this for either of the two variables. However, with an eye towards working with these later on let’s try to avoid as many fractions as possible. The eigenvector is then,
→η\=(η1η2)\=(−7η2η2),η2≠0
η
→
\=
(
η
1
η
2
)
\=
(
−
7
η
2
η
2
)
,
η
2
≠
0
→η(2)\=(−71),using η2\=1
η
→
(
2
)
\=
(
−
7
1
)
,
using
η
2
\=
1
Summarizing we have,
λ1\=−5→η(1)\=(−11)λ2\=1→η(2)\=(−71)
λ
1
\=
−
5
η
→
(
1
)
\=
(
−
1
1
)
λ
2
\=
1
η
→
(
2
)
\=
(
−
7
1
)
Note that the two eigenvectors are linearly independent as predicted.
Example 2 Find the eigenvalues and eigenvectors of the following matrix.
A\=(1−149−13)
A
\=
(
1
−
1
4
9
−
1
3
)
Show Solution
This matrix has fractions in it. That’s life so don’t get excited about it. First, we need the eigenvalues.
det(A−λI)\=∣∣∣1−λ−149−13−λ∣∣∣\=(1−λ)(−13−λ)\+49\=λ2−23λ\+19\=(λ−13)2⇒λ1,2\=13
det
(
A
−
λ
I
)
\=
\|
1
−
λ
−
1
4
9
−
1
3
−
λ
\|
\=
(
1
−
λ
)
(
−
1
3
−
λ
)
\+
4
9
\=
λ
2
−
2
3
λ
\+
1
9
\=
(
λ
−
1
3
)
2
⇒
λ
1
,
2
\=
1
3
So, it looks like we’ve got an eigenvalue of multiplicity 2 here. Remember that the power on the term will be the multiplicity.
Now, let’s find the eigenvector(s). This one is going to be a little different from the first example. There is only one eigenvalue so let’s do the work for that one. We will need to solve the following system,
(23−149−23)(η1η2)\=(00)⇒R1\=32R2
(
2
3
−
1
4
9
−
2
3
)
(
η
1
η
2
)
\=
(
0
0
)
⇒
R
1
\=
3
2
R
2
So, the rows are multiples of each other. We’ll work with the first equation in this example to find the eigenvector.
23η1−η2\=0η2\=23η1
2
3
η
1
−
η
2
\=
0
η
2
\=
2
3
η
1
Recall in the last example we decided that we wanted to make these as “nice” as possible and so should avoid fractions if we can. Sometimes, as in this case, we simply can’t so we’ll have to deal with it. In this case the eigenvector will be,
→η\=(η1η2)\=(η123η1),η1≠0
η
→
\=
(
η
1
η
2
)
\=
(
η
1
2
3
η
1
)
,
η
1
≠
0
→η(1)\=(32),η1\=3
η
→
(
1
)
\=
(
3
2
)
,
η
1
\=
3
Note that by careful choice of the variable in this case we were able to get rid of the fraction that we had. This is something that in general doesn’t much matter if we do or not. However, when we get back to differential equations it will be easier on us if we don’t have any fractions so we will usually try to eliminate them at this step.
Also, in this case we are only going to get a single (linearly independent) eigenvector. We can get other eigenvectors, by choosing different values of η1 η 1. However, each of these will be linearly dependent with the first eigenvector. If you’re not convinced of this try it. Pick some values for η1 η 1 and get a different vector and check to see if the two are linearly dependent.
Recall from the fact above that an eigenvalue of multiplicity k k will have anywhere from 1 to k k linearly independent eigenvectors. In this case we got one. For most of the 2×2 2 × 2 matrices that we’ll be working with this will be the case, although it doesn’t have to be. We can, on occasion, get two.
Example 3 Find the eigenvalues and eigenvectors of the following matrix.
A\=(−4−1722)
A
\=
(
−
4
−
17
2
2
)
Show Solution
So, we’ll start with the eigenvalues.
det(A−λI)\=∣∣∣−4−λ−1722−λ∣∣∣\=(−4−λ)(2−λ)\+34\=λ2\+2λ\+26
det
(
A
−
λ
I
)
\=
\|
−
4
−
λ
−
17
2
2
−
λ
\|
\=
(
−
4
−
λ
)
(
2
−
λ
)
\+
34
\=
λ
2
\+
2
λ
\+
26
This doesn’t factor, so upon using the quadratic formula we arrive at,
λ1,2\=−1±5i
λ
1
,
2
\=
−
1
±
5
i
In this case we get complex eigenvalues which are definitely a fact of life with eigenvalue/eigenvector problems so get used to them.
Finding eigenvectors for complex eigenvalues is identical to the previous two examples, but it will be somewhat messier. So, let’s do that.
λ1\=−1\+5i λ 1 \= − 1 \+ 5 i :
The system that we need to solve this time is
(−4−(−1\+5i)−1722−(−1\+5i))(η1η2)\=(00)(−3−5i−1723−5i)(η1η2)\=(00)
(
−
4
−
(
−
1
\+
5
i
)
−
17
2
2
−
(
−
1
\+
5
i
)
)
(
η
1
η
2
)
\=
(
0
0
)
(
−
3
−
5
i
−
17
2
3
−
5
i
)
(
η
1
η
2
)
\=
(
0
0
)
Now, it’s not super clear that the rows are multiples of each other, but they are. In this case we have,
R1\=−12(3\+5i)R2
R
1
\=
−
1
2
(
3
\+
5
i
)
R
2
This is not something that you need to worry about, we just wanted to make the point. For the work that we’ll be doing later on with differential equations we will just assume that we’ve done everything correctly and we’ve got two rows that are multiples of each other. Therefore, all that we need to do here is pick one of the rows and work with it.
We’ll work with the second row this time.
2η1\+(3−5i)η2\=0
2
η
1
\+
(
3
−
5
i
)
η
2
\=
0
Now we can solve for either of the two variables. However, again looking forward to differential equations, we are going to need the “i i” in the numerator so solve the equation in such a way as this will happen. Doing this gives,
2η1\=−(3−5i)η2η1\=−12(3−5i)η2
2
η
1
\=
−
(
3
−
5
i
)
η
2
η
1
\=
−
1
2
(
3
−
5
i
)
η
2
So, the eigenvector in this case is
→η\=(η1η2)\=(−12(3−5i)η2η2),η2≠0
η
→
\=
(
η
1
η
2
)
\=
(
−
1
2
(
3
−
5
i
)
η
2
η
2
)
,
η
2
≠
0
→η(1)\=(−3\+5i2),η2\=2
η
→
(
1
)
\=
(
−
3
\+
5
i
2
)
,
η
2
\=
2
As with the previous example we choose the value of the variable to clear out the fraction.
Now, the work for the second eigenvector is almost identical and so we’ll not dwell on that too much.
λ2\=−1−5i λ 2 \= − 1 − 5 i :
The system that we need to solve here is
(−4−(−1−5i)−1722−(−1−5i))(η1η2)\=(00)(−3\+5i−1723\+5i)(η1η2)\=(00)
(
−
4
−
(
−
1
−
5
i
)
−
17
2
2
−
(
−
1
−
5
i
)
)
(
η
1
η
2
)
\=
(
0
0
)
(
−
3
\+
5
i
−
17
2
3
\+
5
i
)
(
η
1
η
2
)
\=
(
0
0
)
Working with the second row again gives,
2η1\+(3\+5i)η2\=0⇒η1\=−12(3\+5i)η2
2
η
1
\+
(
3
\+
5
i
)
η
2
\=
0
⇒
η
1
\=
−
1
2
(
3
\+
5
i
)
η
2
The eigenvector in this case is
→η\=(η1η2)\=(−12(3\+5i)η2η2),η2≠0
η
→
\=
(
η
1
η
2
)
\=
(
−
1
2
(
3
\+
5
i
)
η
2
η
2
)
,
η
2
≠
0
→η(2)\=(−3−5i2),η2\=2
η
→
(
2
)
\=
(
−
3
−
5
i
2
)
,
η
2
\=
2
Summarizing,
λ1\=−1\+5i→η(1)\=(−3\+5i2)λ2\=−1−5i→η(2)\=(−3−5i2)
λ
1
\=
−
1
\+
5
i
η
→
(
1
)
\=
(
−
3
\+
5
i
2
)
λ
2
\=
−
1
−
5
i
η
→
(
2
)
\=
(
−
3
−
5
i
2
)
There is a nice fact that we can use to simplify the work when we get complex eigenvalues. We need a bit of terminology first however.
If we start with a complex number,
z\=a\+bi
z
\=
a
\+
b
i
then the **complex conjugate** of z z is
¯¯¯z\=a−bi
z
¯
\=
a
−
b
i
To compute the complex conjugate of a complex number we simply change the sign on the term that contains the “i i”. The complex conjugate of a vector is just the conjugate of each of the vector’s components.
We now have the following fact about complex eigenvalues and eigenvectors.
#### Fact
If A A is an n×n n × n matrix with only real numbers and if λ1\=a\+bi λ 1 \= a \+ b i is an eigenvalue with eigenvector →η(1) η → ( 1 ). Then λ2\=¯¯¯¯¯¯λ1\=a−bi λ 2 \= λ 1 ¯ \= a − b i is also an eigenvalue and its eigenvector is the conjugate of →η(1) η → ( 1 ).
This fact is something that you should feel free to use as you need to in our work.
Now, we need to work one final eigenvalue/eigenvector problem. To this point we’ve only worked with 2×2 2 × 2 matrices and we should work at least one that isn’t 2×2 2 × 2. Also, we need to work one in which we get an eigenvalue of multiplicity greater than one that has more than one linearly independent eigenvector.
Example 4 Find the eigenvalues and eigenvectors of the following matrix.
A\=⎛⎜⎝011101110⎞⎟⎠
A
\=
(
0
1
1
1
0
1
1
1
0
)
Show Solution
Despite the fact that this is a 3×3 3 × 3 matrix, it still works the same as the 2×2 2 × 2 matrices that we’ve been working with. So, start with the eigenvalues
det(A−λI)\=∣∣ ∣∣−λ111−λ111−λ∣∣ ∣∣\=−λ3\+3λ\+2\=(λ−2)(λ\+1)2λ1\=2,λ2,3\=−1
det
(
A
−
λ
I
)
\=
\|
−
λ
1
1
1
−
λ
1
1
1
−
λ
\|
\=
−
λ
3
\+
3
λ
\+
2
\=
(
λ
−
2
)
(
λ
\+
1
)
2
λ
1
\=
2
,
λ
2
,
3
\=
−
1
So, we’ve got a simple eigenvalue and an eigenvalue of multiplicity 2. Note that we used the same method of computing the determinant of a 3×3 3 × 3 matrix that we used in the previous section. We just didn’t show the work.
Let’s now get the eigenvectors. We’ll start with the simple eigenvector.
λ1\=2 λ 1 \= 2 :
Here we’ll need to solve,
⎛⎜⎝−2111−2111−2⎞⎟⎠⎛⎜⎝η1η2η3⎞⎟⎠\=⎛⎜⎝000⎞⎟⎠
(
−
2
1
1
1
−
2
1
1
1
−
2
)
(
η
1
η
2
η
3
)
\=
(
0
0
0
)
This time, unlike the 2×2 2 × 2 cases we worked earlier, we actually need to solve the system. So let’s do that.
⎛⎜⎝−21101−21011−20⎞⎟⎠R1↔R2⇒⎛⎜⎝1−210−211011−20⎞⎟⎠R2\+2R1R3−R1⇒⎛⎜⎝1−2100−33003−30⎞⎟⎠
(
−
2
1
1
0
1
−
2
1
0
1
1
−
2
0
)
R
1
↔
R
2
⇒
(
1
−
2
1
0
−
2
1
1
0
1
1
−
2
0
)
R
2
\+
2
R
1
R
3
−
R
1
⇒
(
1
−
2
1
0
0
−
3
3
0
0
3
−
3
0
)
−13R2⇒⎛⎜⎝1−21001−1003−30⎞⎟⎠R3−3R2R1\+2R2⇒⎛⎜⎝10−1001−100000⎞⎟⎠
−
1
3
R
2
⇒
(
1
−
2
1
0
0
1
−
1
0
0
3
−
3
0
)
R
3
−
3
R
2
R
1
\+
2
R
2
⇒
(
1
0
−
1
0
0
1
−
1
0
0
0
0
0
)
Going back to equations gives,
η1−η3\=0⇒η1\=η3η2−η3\=0⇒η2\=η3
η
1
−
η
3
\=
0
⇒
η
1
\=
η
3
η
2
−
η
3
\=
0
⇒
η
2
\=
η
3
So, again we get infinitely many solutions as we should for eigenvectors. The eigenvector is then,
→η\=⎛⎜⎝η1η2η3⎞⎟⎠\=⎛⎜⎝η3η3η3⎞⎟⎠,η3≠0
η
→
\=
(
η
1
η
2
η
3
)
\=
(
η
3
η
3
η
3
)
,
η
3
≠
0
→η(1)\=⎛⎜⎝111⎞⎟⎠,η3\=1
η
→
(
1
)
\=
(
1
1
1
)
,
η
3
\=
1
Now, let’s do the other eigenvalue.
λ2\=−1 λ 2 \= − 1 :
Here we’ll need to solve,
⎛⎜⎝111111111⎞⎟⎠⎛⎜⎝η1η2η3⎞⎟⎠\=⎛⎜⎝000⎞⎟⎠
(
1
1
1
1
1
1
1
1
1
)
(
η
1
η
2
η
3
)
\=
(
0
0
0
)
Okay, in this case is clear that all three rows are the same and so there isn’t any reason to actually solve the system since we can clear out the bottom two rows to all zeroes in one step. The equation that we get then is,
η1\+η2\+η3\=0⇒η1\=−η2−η3
η
1
\+
η
2
\+
η
3
\=
0
⇒
η
1
\=
−
η
2
−
η
3
So, in this case we get to pick two of the values for free and will still get infinitely many solutions. Here is the general eigenvector for this case,
→η\=⎛⎜⎝η1η2η3⎞⎟⎠\=⎛⎜⎝−η2−η3η2η3⎞⎟⎠,η2≠0 and η3≠0 at the same time
η
→
\=
(
η
1
η
2
η
3
)
\=
(
−
η
2
−
η
3
η
2
η
3
)
,
η
2
≠
0
and
η
3
≠
0
at the same time
Notice the restriction this time. Recall that we only require that the eigenvector not be the zero vector. This means that we can allow one or the other of the two variables to be zero, we just can’t allow both of them to be zero at the same time\!
What this means for us is that we are going to get two linearly independent eigenvectors this time. Here they are.
→η(2)\=⎛⎜⎝−101⎞⎟⎠η2\=0 and η3\=1
η
→
(
2
)
\=
(
−
1
0
1
)
η
2
\=
0
and
η
3
\=
1
→η(3)\=⎛⎜⎝−110⎞⎟⎠η2\=1 and η3\=0
η
→
(
3
)
\=
(
−
1
1
0
)
η
2
\=
1
and
η
3
\=
0
Now when we talked about linear independent vectors in the last [section](https://tutorial.math.lamar.edu/classes/de/LA_Matrix.aspx#LILD) we only looked at n n vectors each with n n components. We can still talk about linear independence in this case however. Recall [back](https://tutorial.math.lamar.edu/classes/de/Wronskian.aspx) with we did linear independence for functions we saw at the time that if two functions were linearly dependent then they were multiples of each other. Well the same thing holds true for vectors. Two vectors will be linearly dependent if they are multiples of each other. In this case there is no way to get →η(2) η → ( 2 ) by multiplying →η(3) η → ( 3 ) by a constant. Therefore, these two vectors must be linearly independent.
So, summarizing up, here are the eigenvalues and eigenvectors for this matrix
λ1\=2→η(1)\=⎛⎜⎝111⎞⎟⎠λ2\=−1→η(2)\=⎛⎜⎝−101⎞⎟⎠λ3\=−1→η(3)\=⎛⎜⎝−110⎞⎟⎠
λ
1
\=
2
η
→
(
1
)
\=
(
1
1
1
)
λ
2
\=
−
1
η
→
(
2
)
\=
(
−
1
0
1
)
λ
3
\=
−
1
η
→
(
3
)
\=
(
−
1
1
0
)
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### Section 5.3 : Review : Eigenvalues & Eigenvectors
If you get nothing out of this quick review of linear algebra you must get this section. Without this section you will not be able to do any of the differential equations work that is in this chapter.
So, let’s start with the following. If we multiply an n × n matrix by an n × 1 vector we will get a new n × 1 vector back. In other words,
A η → \= y →
What we want to know is if it is possible for the following to happen. Instead of just getting a brand new vector out of the multiplication is it possible instead to get the following,
(1) A η → \= λ η →
In other words, is it possible, at least for certain λ and η →, to have matrix multiplication be the same as just multiplying the vector by a constant? Of course, we probably wouldn’t be talking about this if the answer was no. So, it is possible for this to happen, however, it won’t happen for just any value of λ or η →. If we do happen to have a λ and η → for which this works (and they will always come in pairs) then we call λ an **eigenvalue** of A and η → an **eigenvector** of A.
So, how do we go about finding the eigenvalues and eigenvectors for a matrix? Well first notice that if η → \= 0 → then (1) is going to be true for any value of λ and so we are going to make the assumption that η → ≠ 0 →. With that out of the way let’s rewrite (1) a little.
A η → − λ η → \= 0 → A η → − λ I n η → \= 0 → ( A − λ I n ) η → \= 0 →
Notice that before we factored out the η → we added in the appropriately sized identity matrix. This is equivalent to multiplying things by a one and so doesn’t change the value of anything. We needed to do this because without it we would have had the difference of a matrix, A, and a constant, λ, and this can’t be done. We now have the difference of two matrices of the same size which can be done.
So, with this rewrite we see that
(2) ( A − λ I n ) η → \= 0 →
is equivalent to (1). In order to find the eigenvectors for a matrix we will need to solve a homogeneous system. Recall the [fact](https://tutorial.math.lamar.edu/classes/de/LA_Matrix.aspx#System_Fact_2) from the previous section that we know that we will either have exactly one solution (η → \= 0 →) or we will have infinitely many nonzero solutions. Since we’ve already said that we don’t want η → \= 0 → this means that we want the second case.
Knowing this will allow us to find the eigenvalues for a matrix. Recall from this fact that we will get the second case only if the matrix in the system is singular. Therefore, we will need to determine the values of λ for which we get,
det ( A − λ I ) \= 0
Once we have the eigenvalues we can then go back and determine the eigenvectors for each eigenvalue. Let’s take a look at a couple of quick facts about eigenvalues and eigenvectors.
#### Fact
If A is an n × n matrix then det ( A − λ I ) \= 0 is an n th degree polynomial. This polynomial is called the **characteristic polynomial**.
To find eigenvalues of a matrix all we need to do is solve a polynomial. That’s generally not too bad provided we keep n small. Likewise this fact also tells us that for an n × n matrix, A, we will have n eigenvalues if we include all repeated eigenvalues.
#### Fact
If λ 1 , λ 2 , … , λ n is the complete list of eigenvalues for A (including all repeated eigenvalues) then,
1. If
λ
occurs only once in the list then we call
λ
**simple**.
2. If
λ
occurs
k
\>
1
times in the list then we say that
λ
has **multiplicity k**.
3. If
λ
1
,
λ
2
,
…
,
λ
k
(
k
≤
n
) are the simple eigenvalues in the list with corresponding eigenvectors
η
→
(
1
)
,
η
→
(
2
)
, …,
η
→
(
k
)
then the eigenvectors are all linearly independent.
4. If
λ
is an eigenvalue of multiplicity
k
\>
1
then
λ
will have anywhere from 1 to
k
linearly independent eigenvectors.
The usefulness of these facts will become apparent when we get back into differential equations since in that work we will want linearly independent solutions.
Let’s work a couple of examples now to see how we actually go about finding eigenvalues and eigenvectors.
Example 1 Find the eigenvalues and eigenvectors of the following matrix. A \= ( 2 7 − 1 − 6 )
Show Solution
The first thing that we need to do is find the eigenvalues. That means we need the following matrix,
A − λ I \= ( 2 7 − 1 − 6 ) − λ ( 1 0 0 1 ) \= ( 2 − λ 7 − 1 − 6 − λ )
In particular we need to determine where the determinant of this matrix is zero.
det ( A − λ I ) \= ( 2 − λ ) ( − 6 − λ ) \+ 7 \= λ 2 \+ 4 λ − 5 \= ( λ \+ 5 ) ( λ − 1 )
So, it looks like we will have two simple eigenvalues for this matrix, λ 1 \= − 5 and λ 2 \= 1. We will now need to find the eigenvectors for each of these. Also note that according to the fact above, the two eigenvectors should be linearly independent.
To find the eigenvectors we simply plug in each eigenvalue into and solve. So, let’s do that.
λ 1 \= − 5 :
In this case we need to solve the following system.
( 7 7 − 1 − 1 ) η → \= ( 0 0 )
Recall that officially to solve this system we use the following augmented matrix.
( 7 7 0 − 1 − 1 0 ) 1 7 R 1 \+ R 2 ⇒ ( 7 7 0 0 0 0 )
Upon reducing down we see that we get a single equation
7 η 1 \+ 7 η 2 \= 0 ⇒ η 1 \= − η 2
that will yield an infinite number of solutions. This is expected behavior. Recall that we picked the eigenvalues so that the matrix would be singular and so we would get infinitely many solutions.
Notice as well that we could have identified this from the original system. This won’t always be the case, but in the 2 × 2 case we can see from the system that one row will be a multiple of the other and so we will get infinite solutions. From this point on we won’t be actually solving systems in these cases. We will just go straight to the equation and we can use either of the two rows for this equation.
Now, let’s get back to the eigenvector, since that is what we were after. In general then the eigenvector will be any vector that satisfies the following,
η → \= ( η 1 η 2 ) \= ( − η 2 η 2 ) , η 2 ≠ 0
To get this we used the solution to the equation that we found above.
We really don’t want a general eigenvector however so we will pick a value for η 2 to get a specific eigenvector. We can choose anything (except η 2 \= 0), so pick something that will make the eigenvector “nice”. Note as well that since we’ve already assumed that the eigenvector is not zero we must choose a value that will not give us zero, which is why we want to avoid η 2 \= 0 in this case. Here’s the eigenvector for this eigenvalue.
η → ( 1 ) \= ( − 1 1 ) , using η 2 \= 1
Now we get to do this all over again for the second eigenvalue.
λ 2 \= 1 :
We’ll do much less work with this part than we did with the previous part. We will need to solve the following system.
( 1 7 − 1 − 7 ) η → \= ( 0 0 )
Clearly both rows are multiples of each other and so we will get infinitely many solutions. We can choose to work with either row. We’ll run with the first because to avoid having too many minus signs floating around. Doing this gives us,
η 1 \+ 7 η 2 \= 0 η 1 \= − 7 η 2
Note that we can solve this for either of the two variables. However, with an eye towards working with these later on let’s try to avoid as many fractions as possible. The eigenvector is then,
η → \= ( η 1 η 2 ) \= ( − 7 η 2 η 2 ) , η 2 ≠ 0 η → ( 2 ) \= ( − 7 1 ) , using η 2 \= 1
Summarizing we have,
λ 1 \= − 5 η → ( 1 ) \= ( − 1 1 ) λ 2 \= 1 η → ( 2 ) \= ( − 7 1 )
Note that the two eigenvectors are linearly independent as predicted.
Example 2 Find the eigenvalues and eigenvectors of the following matrix. A \= ( 1 − 1 4 9 − 1 3 )
Show Solution
This matrix has fractions in it. That’s life so don’t get excited about it. First, we need the eigenvalues.
det ( A − λ I ) \= \| 1 − λ − 1 4 9 − 1 3 − λ \| \= ( 1 − λ ) ( − 1 3 − λ ) \+ 4 9 \= λ 2 − 2 3 λ \+ 1 9 \= ( λ − 1 3 ) 2 ⇒ λ 1 , 2 \= 1 3
So, it looks like we’ve got an eigenvalue of multiplicity 2 here. Remember that the power on the term will be the multiplicity.
Now, let’s find the eigenvector(s). This one is going to be a little different from the first example. There is only one eigenvalue so let’s do the work for that one. We will need to solve the following system,
( 2 3 − 1 4 9 − 2 3 ) ( η 1 η 2 ) \= ( 0 0 ) ⇒ R 1 \= 3 2 R 2
So, the rows are multiples of each other. We’ll work with the first equation in this example to find the eigenvector.
2 3 η 1 − η 2 \= 0 η 2 \= 2 3 η 1
Recall in the last example we decided that we wanted to make these as “nice” as possible and so should avoid fractions if we can. Sometimes, as in this case, we simply can’t so we’ll have to deal with it. In this case the eigenvector will be,
η → \= ( η 1 η 2 ) \= ( η 1 2 3 η 1 ) , η 1 ≠ 0 η → ( 1 ) \= ( 3 2 ) , η 1 \= 3
Note that by careful choice of the variable in this case we were able to get rid of the fraction that we had. This is something that in general doesn’t much matter if we do or not. However, when we get back to differential equations it will be easier on us if we don’t have any fractions so we will usually try to eliminate them at this step.
Also, in this case we are only going to get a single (linearly independent) eigenvector. We can get other eigenvectors, by choosing different values of η 1. However, each of these will be linearly dependent with the first eigenvector. If you’re not convinced of this try it. Pick some values for η 1 and get a different vector and check to see if the two are linearly dependent.
Recall from the fact above that an eigenvalue of multiplicity k will have anywhere from 1 to k linearly independent eigenvectors. In this case we got one. For most of the 2 × 2 matrices that we’ll be working with this will be the case, although it doesn’t have to be. We can, on occasion, get two.
Example 3 Find the eigenvalues and eigenvectors of the following matrix. A \= ( − 4 − 17 2 2 )
Show Solution
So, we’ll start with the eigenvalues.
det ( A − λ I ) \= \| − 4 − λ − 17 2 2 − λ \| \= ( − 4 − λ ) ( 2 − λ ) \+ 34 \= λ 2 \+ 2 λ \+ 26
This doesn’t factor, so upon using the quadratic formula we arrive at,
λ 1 , 2 \= − 1 ± 5 i
In this case we get complex eigenvalues which are definitely a fact of life with eigenvalue/eigenvector problems so get used to them.
Finding eigenvectors for complex eigenvalues is identical to the previous two examples, but it will be somewhat messier. So, let’s do that.
λ 1 \= − 1 \+ 5 i :
The system that we need to solve this time is
( − 4 − ( − 1 \+ 5 i ) − 17 2 2 − ( − 1 \+ 5 i ) ) ( η 1 η 2 ) \= ( 0 0 ) ( − 3 − 5 i − 17 2 3 − 5 i ) ( η 1 η 2 ) \= ( 0 0 )
Now, it’s not super clear that the rows are multiples of each other, but they are. In this case we have,
R 1 \= − 1 2 ( 3 \+ 5 i ) R 2
This is not something that you need to worry about, we just wanted to make the point. For the work that we’ll be doing later on with differential equations we will just assume that we’ve done everything correctly and we’ve got two rows that are multiples of each other. Therefore, all that we need to do here is pick one of the rows and work with it.
We’ll work with the second row this time.
2 η 1 \+ ( 3 − 5 i ) η 2 \= 0
Now we can solve for either of the two variables. However, again looking forward to differential equations, we are going to need the “i” in the numerator so solve the equation in such a way as this will happen. Doing this gives,
2 η 1 \= − ( 3 − 5 i ) η 2 η 1 \= − 1 2 ( 3 − 5 i ) η 2
So, the eigenvector in this case is
η → \= ( η 1 η 2 ) \= ( − 1 2 ( 3 − 5 i ) η 2 η 2 ) , η 2 ≠ 0 η → ( 1 ) \= ( − 3 \+ 5 i 2 ) , η 2 \= 2
As with the previous example we choose the value of the variable to clear out the fraction.
Now, the work for the second eigenvector is almost identical and so we’ll not dwell on that too much.
λ 2 \= − 1 − 5 i :
The system that we need to solve here is
( − 4 − ( − 1 − 5 i ) − 17 2 2 − ( − 1 − 5 i ) ) ( η 1 η 2 ) \= ( 0 0 ) ( − 3 \+ 5 i − 17 2 3 \+ 5 i ) ( η 1 η 2 ) \= ( 0 0 )
Working with the second row again gives,
2 η 1 \+ ( 3 \+ 5 i ) η 2 \= 0 ⇒ η 1 \= − 1 2 ( 3 \+ 5 i ) η 2
The eigenvector in this case is
η → \= ( η 1 η 2 ) \= ( − 1 2 ( 3 \+ 5 i ) η 2 η 2 ) , η 2 ≠ 0 η → ( 2 ) \= ( − 3 − 5 i 2 ) , η 2 \= 2
Summarizing,
λ 1 \= − 1 \+ 5 i η → ( 1 ) \= ( − 3 \+ 5 i 2 ) λ 2 \= − 1 − 5 i η → ( 2 ) \= ( − 3 − 5 i 2 )
There is a nice fact that we can use to simplify the work when we get complex eigenvalues. We need a bit of terminology first however.
If we start with a complex number,
z \= a \+ b i
then the **complex conjugate** of z is
z ¯ \= a − b i
To compute the complex conjugate of a complex number we simply change the sign on the term that contains the “i”. The complex conjugate of a vector is just the conjugate of each of the vector’s components.
We now have the following fact about complex eigenvalues and eigenvectors.
#### Fact
If A is an n × n matrix with only real numbers and if λ 1 \= a \+ b i is an eigenvalue with eigenvector η → ( 1 ). Then λ 2 \= λ 1 ¯ \= a − b i is also an eigenvalue and its eigenvector is the conjugate of η → ( 1 ).
This fact is something that you should feel free to use as you need to in our work.
Now, we need to work one final eigenvalue/eigenvector problem. To this point we’ve only worked with 2 × 2 matrices and we should work at least one that isn’t 2 × 2. Also, we need to work one in which we get an eigenvalue of multiplicity greater than one that has more than one linearly independent eigenvector.
Example 4 Find the eigenvalues and eigenvectors of the following matrix. A \= ( 0 1 1 1 0 1 1 1 0 )
Show Solution
Despite the fact that this is a 3 × 3 matrix, it still works the same as the 2 × 2 matrices that we’ve been working with. So, start with the eigenvalues
det ( A − λ I ) \= \| − λ 1 1 1 − λ 1 1 1 − λ \| \= − λ 3 \+ 3 λ \+ 2 \= ( λ − 2 ) ( λ \+ 1 ) 2 λ 1 \= 2 , λ 2 , 3 \= − 1
So, we’ve got a simple eigenvalue and an eigenvalue of multiplicity 2. Note that we used the same method of computing the determinant of a 3 × 3 matrix that we used in the previous section. We just didn’t show the work.
Let’s now get the eigenvectors. We’ll start with the simple eigenvector.
λ 1 \= 2 :
Here we’ll need to solve,
( − 2 1 1 1 − 2 1 1 1 − 2 ) ( η 1 η 2 η 3 ) \= ( 0 0 0 )
This time, unlike the 2 × 2 cases we worked earlier, we actually need to solve the system. So let’s do that.
( − 2 1 1 0 1 − 2 1 0 1 1 − 2 0 ) R 1 ↔ R 2 ⇒ ( 1 − 2 1 0 − 2 1 1 0 1 1 − 2 0 ) R 2 \+ 2 R 1 R 3 − R 1 ⇒ ( 1 − 2 1 0 0 − 3 3 0 0 3 − 3 0 ) − 1 3 R 2 ⇒ ( 1 − 2 1 0 0 1 − 1 0 0 3 − 3 0 ) R 3 − 3 R 2 R 1 \+ 2 R 2 ⇒ ( 1 0 − 1 0 0 1 − 1 0 0 0 0 0 )
Going back to equations gives,
η 1 − η 3 \= 0 ⇒ η 1 \= η 3 η 2 − η 3 \= 0 ⇒ η 2 \= η 3
So, again we get infinitely many solutions as we should for eigenvectors. The eigenvector is then,
η → \= ( η 1 η 2 η 3 ) \= ( η 3 η 3 η 3 ) , η 3 ≠ 0 η → ( 1 ) \= ( 1 1 1 ) , η 3 \= 1
Now, let’s do the other eigenvalue.
λ 2 \= − 1 :
Here we’ll need to solve,
( 1 1 1 1 1 1 1 1 1 ) ( η 1 η 2 η 3 ) \= ( 0 0 0 )
Okay, in this case is clear that all three rows are the same and so there isn’t any reason to actually solve the system since we can clear out the bottom two rows to all zeroes in one step. The equation that we get then is,
η 1 \+ η 2 \+ η 3 \= 0 ⇒ η 1 \= − η 2 − η 3
So, in this case we get to pick two of the values for free and will still get infinitely many solutions. Here is the general eigenvector for this case,
η → \= ( η 1 η 2 η 3 ) \= ( − η 2 − η 3 η 2 η 3 ) , η 2 ≠ 0 and η 3 ≠ 0 at the same time
Notice the restriction this time. Recall that we only require that the eigenvector not be the zero vector. This means that we can allow one or the other of the two variables to be zero, we just can’t allow both of them to be zero at the same time\!
What this means for us is that we are going to get two linearly independent eigenvectors this time. Here they are.
η → ( 2 ) \= ( − 1 0 1 ) η 2 \= 0 and η 3 \= 1 η → ( 3 ) \= ( − 1 1 0 ) η 2 \= 1 and η 3 \= 0
Now when we talked about linear independent vectors in the last [section](https://tutorial.math.lamar.edu/classes/de/LA_Matrix.aspx#LILD) we only looked at n vectors each with n components. We can still talk about linear independence in this case however. Recall [back](https://tutorial.math.lamar.edu/classes/de/Wronskian.aspx) with we did linear independence for functions we saw at the time that if two functions were linearly dependent then they were multiples of each other. Well the same thing holds true for vectors. Two vectors will be linearly dependent if they are multiples of each other. In this case there is no way to get η → ( 2 ) by multiplying η → ( 3 ) by a constant. Therefore, these two vectors must be linearly independent.
So, summarizing up, here are the eigenvalues and eigenvectors for this matrix
λ 1 \= 2 η → ( 1 ) \= ( 1 1 1 ) λ 2 \= − 1 η → ( 2 ) \= ( − 1 0 1 ) λ 3 \= − 1 η → ( 3 ) \= ( − 1 1 0 ) |
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