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| Meta Description | They have many uses ... A simple example is that an eigenvector does not change direction in a transformation ... How do we find that vector? |
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| Boilerpipe Text | They have many uses!
A simple example is that an eigenvector
does not change direction
in a transformation:
How do we find that vector?
The Mathematics Of It
For a square matrix
A
, an Eigenvector and Eigenvalue make this equation true:
Let us see it in action:
Example: For this matrix
−6
3
4
5
an eigenvector is
1
4
with a matching eigenvalue of
6
Let's do some
matrix multiplies
to see if that is true.
Av
gives us:
−6
3
4
5
1
4
=
−6×1+3×4
4×1+5×4
=
6
24
λv
gives us :
6
1
4
=
6
24
Yes they are equal!
So we get
Av
=
λv
as promised.
Notice how we multiply a
matrix
by a
vector
and get the same result as when we multiply a
scalar
(just a number) by that
vector
.
How do we find these
eigen things?
We start by finding the
eigenvalue
. We know this equation must be true:
Av
=
λv
Next we put in an
identity matrix
so we are dealing with matrix-vs-matrix:
Av
=
λIv
Bring all to left hand side:
Av − λIv
=
0
If
v
is non-zero then we can (hopefully) solve for
λ
using just the
determinant
:
| A − λI |
=
0
Let's try that equation on our previous example:
Example: Solve for
λ
Start with
| A − λI |
=
0
|
−6
3
4
5
− λ
1
0
0
1
|
=
0
Which is:
−6−λ
3
4
5−λ
= 0
Calculating that determinant gets:
(−6−λ)(5−λ) − 3×4
=
0
Which simplifies to this
Quadratic Equation
:
λ
2
+ λ − 42
=
0
And
solving it
gets:
λ
=
−7 or 6
And yes, there are
two
possible eigenvalues.
Now we know
eigenvalues
, let us find their matching
eigenvectors
.
Example (continued): Find the Eigenvector for the Eigenvalue
λ
=
6
:
Start with:
Av
=
λv
Put in the values we know:
−6
3
4
5
x
y
= 6
x
y
After multiplying we get these two equations:
−6x + 3y
=
6x
4x + 5y
=
6y
Bringing all to left hand side:
−12x + 3y
=
0
4x − 1y
=
0
Either
equation reveals that
y
=
4x
, so the
eigenvector
is any non-zero multiple of this:
1
4
And we get the solution shown at the top of the page:
−6
3
4
5
1
4
=
−6×1+3×4
4×1+5×4
=
6
24
... and also ...
6
1
4
=
6
24
So
Av
=
λv
, and we have success!
Now it is
your turn
to find the eigenvector for the other eigenvalue of
−7
Why?
What is the purpose of these?
One of the cool things is we can use
matrices
to do
transformations
in space, which is used a lot in computer graphics.
In that case the eigenvector is "the direction that doesn't change direction" !
And the eigenvalue is the scale of the stretch:
1
means no change,
2
means doubling in length,
−1
means pointing backwards along the eigenvector's direction
etc
There are also many applications in physics, etc.
Why "Eigen"
Eigen is a German word meaning "own" or "typical"
"das ist ihnen
eigen
"
is German for
"that is
typical
of them"
Sometimes in English we use the word "characteristic", so an eigenvector can be called a "characteristic vector".
Not Just Two Dimensions
Eigenvectors work perfectly well in 3 and higher dimensions.
Example: find the eigenvalues for this 3x3 matrix:
2
0
0
0
4
5
0
4
3
First calculate
A − λI
:
2
0
0
0
4
5
0
4
3
− λ
1
0
0
0
1
0
0
0
1
=
2−λ
0
0
0
4−λ
5
0
4
3−λ
Now the determinant should equal zero:
2−λ
0
0
0
4−λ
5
0
4
3−λ
= 0
Which is:
(2−λ) [ (4−λ)(3−λ) − 5×4 ]
=
0
This ends up being a cubic equation, but just looking at it here we see one of the roots is
2
(because of 2−λ), and the part inside the square brackets is Quadratic, with
roots
of
−1
and
8
.
So the Eigenvalues are
−1
,
2
and
8
Example (continued): find the Eigenvector that matches the Eigenvalue
−1
Put in the values we know:
2
0
0
0
4
5
0
4
3
x
y
z
= −1
x
y
z
After multiplying we get these equations:
2x
=
−x
4y + 5z
=
−y
4y + 3z
=
−z
Bringing all to left hand side:
3x
=
0
5y + 5z
=
0
4y + 4z
=
0
So
x
=
0
, and
y
=
−z
and so the
eigenvector
is any non-zero multiple of this:
0
1
−1
TEST
Av
:
2
0
0
0
4
5
0
4
3
0
1
−1
=
0
4−5
4−3
=
0
−1
1
And
λv
:
−1
0
1
−1
=
0
−1
1
So
Av
=
λv
, yay!
(You can try your hand at the eigenvalues of
2
and
8
)
Rotation
Back in the 2D world again, this matrix will do a rotation by θ:
cos(θ)
−sin(θ)
sin(θ)
cos(θ)
Example: Rotate by 30°
cos(30°)
=
√3
2
and sin(30°)
=
1
2
, so:
cos(30°)
−sin(30°)
sin(30°)
cos(30°)
=
√3
2
−1
2
1
2
√3
2
But if we
rotate all points
, what is the "direction that doesn't change direction"?
Let us work through the mathematics to find out:
First calculate
A − λI
:
√3
2
−1
2
1
2
√3
2
− λ
1
0
0
1
=
√3
2
−λ
−1
2
1
2
√3
2
−λ
Now the determinant should equal zero:
√3
2
−λ
−1
2
1
2
√3
2
−λ
= 0
Which is:
(
√3
2
−λ)(
√3
2
−λ) − (
−1
2
)(
1
2
)
=
0
Which becomes this Quadratic Equation:
λ
2
− (√3)λ + 1
=
0
Whose roots are:
λ
=
√3
2
±
i
2
The eigenvalues are complex!
I don't know how to show you that on a graph, but we still get a solution.
Eigenvector
So, what is an eigenvector that matches, say, the
√3
2
+
i
2
root?
Start with:
Av
=
λv
Put in the values we know:
√3
2
−1
2
1
2
√3
2
x
y
= (
√3
2
+
i
2
)
x
y
After multiplying we get these two equations:
√3
2
x −
1
2
y
=
√3
2
x +
i
2
x
1
2
x +
√3
2
y
=
√3
2
y +
i
2
y
Which simplify to:
−y
=
i
x
x
=
i
y
And the solution is any non-zero multiple of:
i
1
or
−
i
1
Wow, such a simple answer!
Is this just because we chose 30°? Or does it work for any rotation matrix? I will let you work that out! Try another angle, or better still use "cos(θ)" and "sin(θ)".
Oh, and let us
check
at least one of those solutions:
√3
2
−1
2
1
2
√3
2
i
1
=
i
√3
2
−
1
2
i
2
+
√3
2
Does it match this?
(
√3
2
+
i
2
)
i
1
=
i
√3
2
−
1
2
√3
2
+
i
2
Oh yes it does! |
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Close
# Eigenvector and Eigenvalue
They have many uses\!
A simple example is that an eigenvector **does not change direction** in a transformation:
How do we find that vector?
## The Mathematics Of It
For a square matrix **A**, an Eigenvector and Eigenvalue make this equation true:
Let us see it in action:
Example: For this matrix
−6
3
4
5
an eigenvector is
1
4
with a matching eigenvalue of 6
Let's do some [matrix multiplies](https://www.mathsisfun.com/algebra/matrix-multiplying.html) to see if that is true.
Av gives us:
−6
3
4
5
1
4
\=
−6×1+3×4
4×1+5×4
\=
6
24
λv gives us :
6
1
4
\=
6
24
Yes they are equal\!
So we get Av \= λv as promised.
Notice how we multiply a **matrix** by a **vector** and get the same result as when we multiply a **scalar** (just a number) by that **vector**.
## How do we find these eigen things?
We start by finding the **eigenvalue**. We know this equation must be true:
Av \= λv
Next we put in an [identity matrix](https://www.mathsisfun.com/algebra/matrix-types.html) so we are dealing with matrix-vs-matrix:
Av \= λIv
Bring all to left hand side:
Av − λIv \= 0
If **v** is non-zero then we can (hopefully) solve for λ using just the [determinant](https://www.mathsisfun.com/algebra/matrix-determinant.html):
\| A − λI \| \= 0
Let's try that equation on our previous example:
### Example: Solve for λ
Start with \| A − λI \| \= 0
| | | | |
|---|---|---|---|
| **\|** | −6 3 4 5 − λ 1 0 0 1 | **\|** | \= 0 |
Which is:
−6−λ
3
4
5−λ
\= 0
Calculating that determinant gets:
(−6−λ)(5−λ) − 3×4 \= 0
Which simplifies to this [Quadratic Equation](https://www.mathsisfun.com/algebra/quadratic-equation.html):
λ2 + λ − 42 \= 0
And [solving it](https://www.mathsisfun.com/quadratic-equation-solver.html) gets:
λ \= −7 or 6
And yes, there are **two** possible eigenvalues.
Now we know **eigenvalues**, let us find their matching **eigenvectors**.
### Example (continued): Find the Eigenvector for the Eigenvalue **λ \= 6**:
Start with:
Av \= λv
Put in the values we know:
−6
3
4
5
x
y
\= 6
x
y
After multiplying we get these two equations:
| | | |
|---|---|---|
| −6x + 3y | \= | 6x |
| 4x + 5y | \= | 6y |
Bringing all to left hand side:
| | | |
|---|---|---|
| −12x + 3y | \= | 0 |
| 4x − 1y | \= | 0 |
*Either* equation reveals that **y \= 4x**, so the **eigenvector** is any non-zero multiple of this:
1
4
And we get the solution shown at the top of the page:
−6
3
4
5
1
4
\=
−6×1+3×4
4×1+5×4
\=
6
24
... and also ...
6
1
4
\=
6
24
So Av \= λv, and we have success\!
Now it is **your turn** to find the eigenvector for the other eigenvalue of −7
## Why?
What is the purpose of these?
One of the cool things is we can use [matrices](https://www.mathsisfun.com/algebra/matrix-introduction.html) to do [transformations](https://www.mathsisfun.com/algebra/matrix-transform.html) in space, which is used a lot in computer graphics.
In that case the eigenvector is "the direction that doesn't change direction" \!
And the eigenvalue is the scale of the stretch:
- **1** means no change,
- **2** means doubling in length,
- **−1** means pointing backwards along the eigenvector's direction
- etc
There are also many applications in physics, etc.
## Why "Eigen"
Eigen is a German word meaning "own" or "typical"
*"das ist ihnen **eigen**"* is German for *"that is **typical** of them"*
Sometimes in English we use the word "characteristic", so an eigenvector can be called a "characteristic vector".
## Not Just Two Dimensions
Eigenvectors work perfectly well in 3 and higher dimensions.
Example: find the eigenvalues for this 3x3 matrix:
2
0
0
0
4
5
0
4
3
First calculate A − λI:
2
0
0
0
4
5
0
4
3
− λ
1
0
0
0
1
0
0
0
1
\=
2−λ
0
0
0
4−λ
5
0
4
3−λ
Now the determinant should equal zero:
2−λ
0
0
0
4−λ
5
0
4
3−λ
\= 0
Which is:
(2−λ) \[ (4−λ)(3−λ) − 5×4 \] \= 0
This ends up being a cubic equation, but just looking at it here we see one of the roots is **2** (because of 2−λ), and the part inside the square brackets is Quadratic, with [roots](https://www.mathsisfun.com/quadratic-equation-solver.html) of **−1** and **8**.
So the Eigenvalues are **−1**, **2** and **8**
### Example (continued): find the Eigenvector that matches the Eigenvalue **−1**
Put in the values we know:
2
0
0
0
4
5
0
4
3
x
y
z
\= −1
x
y
z
After multiplying we get these equations:
| | | |
|---|---|---|
| 2x | \= | −x |
| 4y + 5z | \= | −y |
| 4y + 3z | \= | −z |
Bringing all to left hand side:
| | | |
|---|---|---|
| 3x | \= | 0 |
| 5y + 5z | \= | 0 |
| 4y + 4z | \= | 0 |
So **x \= 0**, and **y \= −z** and so the **eigenvector** is any non-zero multiple of this:
0
1
−1
TEST Av:
2
0
0
0
4
5
0
4
3
0
1
−1
\=
0
4−5
4−3
\=
0
−1
1
And λv:
−1
0
1
−1
\=
0
−1
1
So Av \= λv, yay\!
(You can try your hand at the eigenvalues of **2** and **8**)
## Rotation
Back in the 2D world again, this matrix will do a rotation by θ:
cos(θ)
−sin(θ)
sin(θ)
cos(θ)
### Example: Rotate by 30°
cos(30°) \= *√3***2** and sin(30°) \= *1***2**, so:
cos(30°)
−sin(30°)
sin(30°)
cos(30°)
\=
*√3***2**
*−1***2**
*1***2**
*√3***2**
But if we **rotate all points**, what is the "direction that doesn't change direction"?
Let us work through the mathematics to find out:
First calculate A − λI:
*√3***2**
*−1***2**
*1***2**
*√3***2**
− λ
1
0
0
1
\=
*√3***2**−λ
*−1***2**
*1***2**
*√3***2**−λ
Now the determinant should equal zero:
*√3***2**−λ
*−1***2**
*1***2**
*√3***2**−λ
\= 0
Which is:
(*√3***2**−λ)(*√3***2**−λ) − (*−1***2**)(*1***2**) \= 0
Which becomes this Quadratic Equation:
λ2 − (√3)λ + 1 \= 0
Whose roots are:
λ \= *√3***2** ± ****i******2**
The eigenvalues are complex\!
I don't know how to show you that on a graph, but we still get a solution.
### Eigenvector
So, what is an eigenvector that matches, say, the *√3***2** + ****i******2** root?
Start with:
Av \= λv
Put in the values we know:
*√3***2**
*−1***2**
*1***2**
*√3***2**
x
y
\= (*√3***2** + ****i******2**)
x
y
After multiplying we get these two equations:
*√3***2**x − *1***2**y \= *√3***2**x + ****i******2**x
*1***2**x + *√3***2**y \= *√3***2**y + ****i******2**y
Which simplify to:
−y \= ***i***x
x \= ***i***y
And the solution is any non-zero multiple of:
***i***
1
or
−***i***
1
**Wow, such a simple answer\!**
*Is this just because we chose 30°? Or does it work for any rotation matrix? I will let you work that out! Try another angle, or better still use "cos(θ)" and "sin(θ)".*
Oh, and let us **check** at least one of those solutions:
*√3***2**
*−1***2**
*1***2**
*√3***2**
***i***
1
\=
***i****√3***2** − *1***2**
****i******2** + *√3***2**
Does it match this?
(*√3***2** + ****i******2**)
***i***
1
\=
***i****√3***2** − *1***2**
*√3***2** + ****i******2**
Oh yes it does\!
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| Readable Markdown | They have many uses\!
A simple example is that an eigenvector **does not change direction** in a transformation:
How do we find that vector?
## The Mathematics Of It
For a square matrix **A**, an Eigenvector and Eigenvalue make this equation true:
Let us see it in action:
Example: For this matrix
−6
3
4
5
an eigenvector is
1
4
with a matching eigenvalue of 6
Let's do some [matrix multiplies](https://www.mathsisfun.com/algebra/matrix-multiplying.html) to see if that is true.
Av gives us:
−6
3
4
5
1
4
\=
−6×1+3×4
4×1+5×4
\=
6
24
λv gives us :
6
1
4
\=
6
24
Yes they are equal\!
So we get Av \= λv as promised.
Notice how we multiply a **matrix** by a **vector** and get the same result as when we multiply a **scalar** (just a number) by that **vector**.
## How do we find these eigen things?
We start by finding the **eigenvalue**. We know this equation must be true:
Av \= λv
Next we put in an [identity matrix](https://www.mathsisfun.com/algebra/matrix-types.html) so we are dealing with matrix-vs-matrix:
Av \= λIv
Bring all to left hand side:
Av − λIv \= 0
If **v** is non-zero then we can (hopefully) solve for λ using just the [determinant](https://www.mathsisfun.com/algebra/matrix-determinant.html):
\| A − λI \| \= 0
Let's try that equation on our previous example:
### Example: Solve for λ
Start with \| A − λI \| \= 0
| | | | |
|---|---|---|---|
| **\|** | −6 3 4 5 − λ 1 0 0 1 | **\|** | \= 0 |
Which is:
−6−λ
3
4
5−λ
\= 0
Calculating that determinant gets:
(−6−λ)(5−λ) − 3×4 \= 0
Which simplifies to this [Quadratic Equation](https://www.mathsisfun.com/algebra/quadratic-equation.html):
λ2 + λ − 42 \= 0
And [solving it](https://www.mathsisfun.com/quadratic-equation-solver.html) gets:
λ \= −7 or 6
And yes, there are **two** possible eigenvalues.
Now we know **eigenvalues**, let us find their matching **eigenvectors**.
### Example (continued): Find the Eigenvector for the Eigenvalue **λ \= 6**:
Start with:
Av \= λv
Put in the values we know:
−6
3
4
5
x
y
\= 6
x
y
After multiplying we get these two equations:
| | | |
|---|---|---|
| −6x + 3y | \= | 6x |
| 4x + 5y | \= | 6y |
Bringing all to left hand side:
| | | |
|---|---|---|
| −12x + 3y | \= | 0 |
| 4x − 1y | \= | 0 |
*Either* equation reveals that **y \= 4x**, so the **eigenvector** is any non-zero multiple of this:
1
4
And we get the solution shown at the top of the page:
−6
3
4
5
1
4
\=
−6×1+3×4
4×1+5×4
\=
6
24
... and also ...
6
1
4
\=
6
24
So Av \= λv, and we have success\!
Now it is **your turn** to find the eigenvector for the other eigenvalue of −7
## Why?
What is the purpose of these?
One of the cool things is we can use [matrices](https://www.mathsisfun.com/algebra/matrix-introduction.html) to do [transformations](https://www.mathsisfun.com/algebra/matrix-transform.html) in space, which is used a lot in computer graphics.
In that case the eigenvector is "the direction that doesn't change direction" \!
And the eigenvalue is the scale of the stretch:
- **1** means no change,
- **2** means doubling in length,
- **−1** means pointing backwards along the eigenvector's direction
- etc
There are also many applications in physics, etc.
## Why "Eigen"
Eigen is a German word meaning "own" or "typical"
*"das ist ihnen **eigen**"* is German for *"that is **typical** of them"*
Sometimes in English we use the word "characteristic", so an eigenvector can be called a "characteristic vector".
## Not Just Two Dimensions
Eigenvectors work perfectly well in 3 and higher dimensions.
Example: find the eigenvalues for this 3x3 matrix:
2
0
0
0
4
5
0
4
3
First calculate A − λI:
2
0
0
0
4
5
0
4
3
− λ
1
0
0
0
1
0
0
0
1
\=
2−λ
0
0
0
4−λ
5
0
4
3−λ
Now the determinant should equal zero:
2−λ
0
0
0
4−λ
5
0
4
3−λ
\= 0
Which is:
(2−λ) \[ (4−λ)(3−λ) − 5×4 \] \= 0
This ends up being a cubic equation, but just looking at it here we see one of the roots is **2** (because of 2−λ), and the part inside the square brackets is Quadratic, with [roots](https://www.mathsisfun.com/quadratic-equation-solver.html) of **−1** and **8**.
So the Eigenvalues are **−1**, **2** and **8**
### Example (continued): find the Eigenvector that matches the Eigenvalue **−1**
Put in the values we know:
2
0
0
0
4
5
0
4
3
x
y
z
\= −1
x
y
z
After multiplying we get these equations:
| | | |
|---|---|---|
| 2x | \= | −x |
| 4y + 5z | \= | −y |
| 4y + 3z | \= | −z |
Bringing all to left hand side:
| | | |
|---|---|---|
| 3x | \= | 0 |
| 5y + 5z | \= | 0 |
| 4y + 4z | \= | 0 |
So **x \= 0**, and **y \= −z** and so the **eigenvector** is any non-zero multiple of this:
0
1
−1
TEST Av:
2
0
0
0
4
5
0
4
3
0
1
−1
\=
0
4−5
4−3
\=
0
−1
1
And λv:
−1
0
1
−1
\=
0
−1
1
So Av \= λv, yay\!
(You can try your hand at the eigenvalues of **2** and **8**)
## Rotation
Back in the 2D world again, this matrix will do a rotation by θ:
cos(θ)
−sin(θ)
sin(θ)
cos(θ)
### Example: Rotate by 30°
cos(30°) \= *√3***2** and sin(30°) \= *1***2**, so:
cos(30°)
−sin(30°)
sin(30°)
cos(30°)
\=
*√3***2**
*−1***2**
*1***2**
*√3***2**
But if we **rotate all points**, what is the "direction that doesn't change direction"?
Let us work through the mathematics to find out:
First calculate A − λI:
*√3***2**
*−1***2**
*1***2**
*√3***2**
− λ
1
0
0
1
\=
*√3***2**−λ
*−1***2**
*1***2**
*√3***2**−λ
Now the determinant should equal zero:
*√3***2**−λ
*−1***2**
*1***2**
*√3***2**−λ
\= 0
Which is:
(*√3***2**−λ)(*√3***2**−λ) − (*−1***2**)(*1***2**) \= 0
Which becomes this Quadratic Equation:
λ2 − (√3)λ + 1 \= 0
Whose roots are:
λ \= *√3***2** ± ****i******2**
The eigenvalues are complex\!
I don't know how to show you that on a graph, but we still get a solution.
### Eigenvector
So, what is an eigenvector that matches, say, the *√3***2** + ****i******2** root?
Start with:
Av \= λv
Put in the values we know:
*√3***2**
*−1***2**
*1***2**
*√3***2**
x
y
\= (*√3***2** + ****i******2**)
x
y
After multiplying we get these two equations:
*√3***2**x − *1***2**y \= *√3***2**x + ****i******2**x
*1***2**x + *√3***2**y \= *√3***2**y + ****i******2**y
Which simplify to:
−y \= ***i***x
x \= ***i***y
And the solution is any non-zero multiple of:
***i***
1
or
−***i***
1
**Wow, such a simple answer\!**
*Is this just because we chose 30°? Or does it work for any rotation matrix? I will let you work that out! Try another angle, or better still use "cos(θ)" and "sin(θ)".*
Oh, and let us **check** at least one of those solutions:
*√3***2**
*−1***2**
*1***2**
*√3***2**
***i***
1
\=
***i****√3***2** − *1***2**
****i******2** + *√3***2**
Does it match this?
(*√3***2** + ****i******2**)
***i***
1
\=
***i****√3***2** − *1***2**
*√3***2** + ****i******2**
Oh yes it does\! |
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