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| Meta Title | Eigenvectors of a Matrix Calculator (with Eigenvalues) - Online |
| Meta Description | Tool to calculate eigenvectors of a matrix. Eigenvectors of a matrix are vectors whose direction remains unchanged after multiplying by the matrix. They are associated with an eigenvalue. |
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Eigenvectors of a Matrix
Tool to calculate eigenvectors of a matrix. Eigenvectors of a matrix are vectors whose direction remains unchanged after multiplying by the matrix. They are associated with an eigenvalue.
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Eigenvectors of a Matrix
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Tag(s) :
Matrix
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Mathematics
Matrix
Eigenvectors of a Matrix
Answers to Questions (FAQ)
What are eigen vectors of a matrix? (Definition)
An
eigenvector
of a matrix is a characteristic vector (or privileged axis or direction) on which a linear transformation behaves like a scalar
multiplication
by a constant named
eigenvalue
.
In other words, these are the vectors that only change by one scale when
multiplied
by the matrix.
The set of
eigenvectors
form an
eigenspace
.
How to calculate eigenvectors of a matrix?
To find
eigenvectors
, take $ M $ a square matrix of size $ n $ and $ \lambda_i $ its
eigenvalues
.
Eigenvectors
are the solution of the system $ ( M − \lambda I_n ) \vec{X} = \vec{0} $ with $ I_n $ the identity matrix.
Example:
The 2x2 matrix $$ M=\begin{bmatrix} 1 & 2 \\ 4 & 3 \end{bmatrix} $$
Eigenvalues
for the matrix $ M $ are $ \lambda_1 = 5 $ and $ \lambda_2 = -1 $ (see tool for calculating matrix
eigenvalues
).
For each
eigenvalue
, look for the associated
eigenvector
.
Example:
For $ \lambda_1 = 5 $, solve $ ( M − 5 I_n ) X = \vec{0} $: $$ \begin{bmatrix} 1-5 & 2 \\ 4 & 3-5 \end{bmatrix} . \begin{bmatrix} x_1 \\ x_2 \end{bmatrix} = \begin{bmatrix} 0 \\ 0 \end{bmatrix} $$ be the equivalent
system of equations
$$ \begin{align} -4 x_1 + 2 x_2 &= 0 \\ 4 x_1 - 2 x_2 &= 0 \end{align} $$ which admits several solutions including $$ \begin{array}{c} x_1 = 1 \\ x_2 = 2 \end{array} $$ So the
eigenvector
associated to $ \lambda_1 = 5 $ is $ \vec{v_1} = \begin{bmatrix} 1 \\ 2 \end{bmatrix} $
Example:
For $ \lambda_2 = -1 $, solve $ ( M + I_n ) X = \vec{0} $ like this: $$ \begin{bmatrix} 1+1 & 2 \\ 4 & 3+1 \end{bmatrix} . \begin{bmatrix} x_1 \\ x_2 \end{bmatrix} = \begin{bmatrix} 0 \\ 0 \end{bmatrix} \\ \iff \begin{align} 2 x_1 + 2 x_2 &= 0 \\ 4 x_1 + 4 x_2 &= 0 \end{align} \Rightarrow \begin{array}{c} x_1 = -1 \\ x_2 = 1 \end{array} $$
So the
eigenvector
associated to $ \lambda_2 = -1 $ is $ \vec{v_2} = \begin{bmatrix} -1 \\ 1 \end{bmatrix} $.
Why use the eigenvectors of a matrix?
Eigenvectors
can be used to simplify certain calculations, to understand the linear transformations induced by the matrix and to solve problems related to
eigenvalues
.
How to prove that a matrix is diagonalizable?
A matrix $ M $ matrix of order $ n $ is a
diagonalizable matrix
if it has $ n $
eigenvectors
associated with $ n $ distinct
eigenvalues
.
That is, it has enough linearly independent
eigenvectors
to form a basis for the vector space in which it operates (necessary condition for its
diagonalization
).
Does a zero vector as an eigenvector exist?
The definition of the
eigenvector
precludes its nullity. However, if in a calculation the number of independent
eigenvectors
is less than the number of
eigenvalues
, dCode will sometimes display a null vector.
Source code
dCode retains ownership of the "Eigenvectors of a Matrix" source code. Any algorithm for the "Eigenvectors of a Matrix" algorithm, applet or snippet or script (converter, solver, encryption / decryption, encoding / decoding, ciphering / deciphering, breaker, translator), or any "Eigenvectors of a Matrix" functions (calculate, convert, solve, decrypt / encrypt, decipher / cipher, decode / encode, translate) written in any informatic language (Python, Java, PHP, C#, Javascript, Matlab, etc.) or any database download or API access for "Eigenvectors of a Matrix" or any other element are not public (except explicit open source licence). Same with the download for offline use on PC, mobile, tablet, iPhone or Android app.
Reminder: dCode is an educational and teaching resource, accessible online for free and for everyone.
Cite dCode
The content of the page "Eigenvectors of a Matrix" and its results may be freely copied and reused, including for commercial purposes, provided that dCode.fr is cited as the source (Creative Commons CC-BY free distribution license).
Exporting the results is free and can be done simply by clicking on the export icons ⤓ (.csv or .txt format) or ⧉ (copy and paste).
To cite dCode.fr on another website, use the link:
https://www.dcode.fr/matrix-eigenvectors
In a scientific article or book, the recommended bibliographic citation is:
Eigenvectors of a Matrix
on dCode.fr [online website], retrieved on 2026-01-09,
https://www.dcode.fr/matrix-eigenvectors
Summary
https://www.dcode.fr/matrix-eigenvectors
© 2026
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— The ultimate collection of tools for games, math, and puzzles.
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Search for a tool
Eigenvectors of a Matrix
Tool to calculate eigenvectors of a matrix. Eigenvectors of a matrix are vectors whose direction remains unchanged after multiplying by the matrix. They are associated with an eigenvalue.
Results
Eigenvectors of a Matrix - [dCode](https://www.dcode.fr/)
Tag(s) : Matrix
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# Eigenvectors of a Matrix
1. [Mathematics](https://www.dcode.fr/tools-list#mathematics)
2. [Matrix](https://www.dcode.fr/tools-list#matrix)
3. [Eigenvectors of a Matrix](https://www.dcode.fr/matrix-eigenvectors)
## Eigenvectors Calculator
## Answers to Questions (FAQ)
### What are eigen vectors of a matrix? (Definition)
An eigenvector of a matrix is a characteristic vector (or privileged axis or direction) on which a linear transformation behaves like a scalar [multiplication](https://www.dcode.fr/big-numbers-multiplication) by a constant named [eigenvalue](https://www.dcode.fr/matrix-eigenvalues).
In other words, these are the vectors that only change by one scale when [multiplied](https://www.dcode.fr/big-numbers-multiplication) by the matrix.
The set of eigenvectors form an [eigenspace](https://www.dcode.fr/matrix-eigenspaces).
### How to calculate eigenvectors of a matrix?
To find eigenvectors, take \$ M \$ a square matrix of size \$ n \$ and \$ \\lambda\_i \$ its [eigenvalues](https://www.dcode.fr/matrix-eigenvalues). Eigenvectors are the solution of the system \$ ( M − \\lambda I\_n ) \\vec{X} = \\vec{0} \$ with \$ I\_n \$ the identity matrix.
Example: The 2x2 matrix \$\$ M=\\begin{bmatrix} 1 & 2 \\\\ 4 & 3 \\end{bmatrix} \$\$
[Eigenvalues](https://www.dcode.fr/matrix-eigenvalues) for the matrix \$ M \$ are \$ \\lambda\_1 = 5 \$ and \$ \\lambda\_2 = -1 \$ (see tool for calculating matrix [eigenvalues](https://www.dcode.fr/matrix-eigenvalues)).
For each [eigenvalue](https://www.dcode.fr/matrix-eigenvalues), look for the associated eigenvector.
Example: For \$ \\lambda\_1 = 5 \$, solve \$ ( M − 5 I\_n ) X = \\vec{0} \$: \$\$ \\begin{bmatrix} 1-5 & 2 \\\\ 4 & 3-5 \\end{bmatrix} . \\begin{bmatrix} x\_1 \\\\ x\_2 \\end{bmatrix} = \\begin{bmatrix} 0 \\\\ 0 \\end{bmatrix} \$\$ be the equivalent [system of equations](https://www.dcode.fr/equation-solver) \$\$ \\begin{align} -4 x\_1 + 2 x\_2 &= 0 \\\\ 4 x\_1 - 2 x\_2 &= 0 \\end{align} \$\$ which admits several solutions including \$\$ \\begin{array}{c} x\_1 = 1 \\\\ x\_2 = 2 \\end{array} \$\$ So the eigenvector associated to \$ \\lambda\_1 = 5 \$ is \$ \\vec{v\_1} = \\begin{bmatrix} 1 \\\\ 2 \\end{bmatrix} \$
Example: For \$ \\lambda\_2 = -1 \$, solve \$ ( M + I\_n ) X = \\vec{0} \$ like this: \$\$ \\begin{bmatrix} 1+1 & 2 \\\\ 4 & 3+1 \\end{bmatrix} . \\begin{bmatrix} x\_1 \\\\ x\_2 \\end{bmatrix} = \\begin{bmatrix} 0 \\\\ 0 \\end{bmatrix} \\\\ \\iff \\begin{align} 2 x\_1 + 2 x\_2 &= 0 \\\\ 4 x\_1 + 4 x\_2 &= 0 \\end{align} \\Rightarrow \\begin{array}{c} x\_1 = -1 \\\\ x\_2 = 1 \\end{array} \$\$
So the eigenvector associated to \$ \\lambda\_2 = -1 \$ is \$ \\vec{v\_2} = \\begin{bmatrix} -1 \\\\ 1 \\end{bmatrix} \$.
### Why use the eigenvectors of a matrix?
Eigenvectors can be used to simplify certain calculations, to understand the linear transformations induced by the matrix and to solve problems related to [eigenvalues](https://www.dcode.fr/matrix-eigenvalues).
### How to prove that a matrix is diagonalizable?
A matrix \$ M \$ matrix of order \$ n \$ is a [diagonalizable matrix](https://www.dcode.fr/matrix-diagonalization) if it has \$ n \$ eigenvectors associated with \$ n \$ distinct [eigenvalues](https://www.dcode.fr/matrix-eigenvalues).
That is, it has enough linearly independent eigenvectors to form a basis for the vector space in which it operates (necessary condition for its [diagonalization](https://www.dcode.fr/matrix-diagonalization)).
### Does a zero vector as an eigenvector exist?
The definition of the eigenvector precludes its nullity. However, if in a calculation the number of independent eigenvectors is less than the number of [eigenvalues](https://www.dcode.fr/matrix-eigenvalues), dCode will sometimes display a null vector.
❓ Ask a new question
## Source code
dCode retains ownership of the "Eigenvectors of a Matrix" source code. Any algorithm for the "Eigenvectors of a Matrix" algorithm, applet or snippet or script (converter, solver, encryption / decryption, encoding / decoding, ciphering / deciphering, breaker, translator), or any "Eigenvectors of a Matrix" functions (calculate, convert, solve, decrypt / encrypt, decipher / cipher, decode / encode, translate) written in any informatic language (Python, Java, PHP, C\#, Javascript, Matlab, etc.) or any database download or API access for "Eigenvectors of a Matrix" or any other element are not public (except explicit open source licence). Same with the download for offline use on PC, mobile, tablet, iPhone or Android app.
Reminder: dCode is an educational and teaching resource, accessible online for free and for everyone.
## Cite dCode
The content of the page "Eigenvectors of a Matrix" and its results may be freely copied and reused, including for commercial purposes, provided that dCode.fr is cited as the source (Creative Commons CC-BY free distribution license).
Exporting the results is free and can be done simply by clicking on the export icons ⤓ (.csv or .txt format) or ⧉ (copy and paste).
To cite dCode.fr on another website, use the link: https://www.dcode.fr/matrix-eigenvectors
In a scientific article or book, the recommended bibliographic citation is: *Eigenvectors of a Matrix* on dCode.fr \[online website\], retrieved on 2026-01-09, https://www.dcode.fr/matrix-eigenvectors
[](https://www.dcode.fr/vecteurs-propres-matrice)
Summary
- [Eigenvectors Calculator](https://www.dcode.fr/matrix-eigenvectors#f0)
- [What are eigen vectors of a matrix? (Definition)](https://www.dcode.fr/matrix-eigenvectors#q1)
- [How to calculate eigenvectors of a matrix?](https://www.dcode.fr/matrix-eigenvectors#q2)
- [Why use the eigenvectors of a matrix?](https://www.dcode.fr/matrix-eigenvectors#q3)
- [How to prove that a matrix is diagonalizable?](https://www.dcode.fr/matrix-eigenvectors#q4)
- [Does a zero vector as an eigenvector exist?](https://www.dcode.fr/matrix-eigenvectors#q5)
Similar pages
- [Matrix Diagonalization](https://www.dcode.fr/matrix-diagonalization)
- [Characteristic Polynomial of a Matrix](https://www.dcode.fr/matrix-characteristic-polynomial)
- [Eigenvalues of a Matrix](https://www.dcode.fr/matrix-eigenvalues)
- [Eigenspaces of a Matrix](https://www.dcode.fr/matrix-eigenspaces)
- [Jordan Normal Form Matrix](https://www.dcode.fr/matrix-jordan)
- [Rank of a Matrix](https://www.dcode.fr/matrix-rank)
- [Schur Decomposition (Matrix)](https://www.dcode.fr/matrix-schur-decomposition)
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