Search for a tool
Eigenspaces of a Matrix

Tool to calculate eigenspaces associated to eigenvalues of any size matrix (also called vectorial spaces Vect).

Results

Eigenspaces of a Matrix -

Tag(s) : Matrix

Share
dCode and more

dCode is free and its tools are a valuable help in games, maths, geocaching, puzzles and problems to solve every day!
A suggestion ? a feedback ? a bug ? an idea ? Write to dCode!

Please, check our dCode Discord community for help requests!
NB: for encrypted messages, test our automatic cipher identifier!

Feedback and suggestions are welcome so that dCode offers the best 'Eigenspaces of a Matrix' tool for free! Thank you!

# Eigenspaces of a Matrix

## Eigenvectors Calculator

### What is an eigenspace of an eigen value of a matrix? (Definition)

For a matrix $M$ having for eigenvalues $\lambda_i$, an eigenspace $E$ associated with an eigenvalue $\lambda_i$ is the set (the basis) of eigenvectors $\vec{v_i}$ which have the same eigenvalue.

That is to say the kernel (or nullspace) of $M - I \lambda_i$.

### How to calculate the eigenspaces associated with an eigenvalue?

For an eigenvalue $\lambda_i$, calculate the matrix $M - I \lambda_i$ (with I the identity matrix) (also works by calculating $I \lambda_i - M$) and calculate for which set of vector $\vec{v}$, the product of my matrix by the vector is equal to the null vector $\vec{0}$

Example: The 2x2 matrix $M = \begin{bmatrix} -1 & 2 \\ 2 & -1 \end{bmatrix}$ has eigenvalues $\lambda_1 = -3$ and $\lambda_2 = 1$, the computation of the proper set associated with $\lambda_1$ is $\begin{bmatrix} -1 + 3 & 2 \\ 2 & -1 + 3 \end{bmatrix} . \begin{bmatrix} v_1 \\ v_2 \end{bmatrix} = \begin{bmatrix} 0 \\ 0 \end{bmatrix}$ which has for solution $v_1 = -v_2$. The eigenspace $E_{\lambda_1}$ is therefore the set of vectors $\begin{bmatrix} v_1 \\ v_2 \end{bmatrix}$ of the form $a \begin{bmatrix} -1 \\ 1 \end{bmatrix} , a \in \mathbb{R}$. The vector space is written $\text{Vect} \left\{ \begin{pmatrix} -1 \\ 1 \end{pmatrix} \right\}$

### How to build a basis for an eigenspace?

To construct a basis for an eigenspace associated with its eigenvalues $\lambda$ and its corresponding eigenvectors $\vec{v}$, select a linearly independent set of these vectors.

This set of linearly independent vectors forms a basis of the eigenspace.

## Source code

dCode retains ownership of the "Eigenspaces of a Matrix" source code. Except explicit open source licence (indicated Creative Commons / free), the "Eigenspaces of a Matrix" algorithm, the applet or snippet (converter, solver, encryption / decryption, encoding / decoding, ciphering / deciphering, breaker, translator), or the "Eigenspaces 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.) and all data download, script, or API access for "Eigenspaces of a Matrix" are not public, same for offline use on PC, mobile, tablet, iPhone or Android app!
Reminder : dCode is free to use.

## Cite dCode

The copy-paste of the page "Eigenspaces of a Matrix" or any of its results, is allowed (even for commercial purposes) as long as you credit dCode!
Exporting results as a .csv or .txt file is free by clicking on the export icon
Cite as source (bibliography):
Eigenspaces of a Matrix on dCode.fr [online website], retrieved on 2024-06-25, https://www.dcode.fr/matrix-eigenspaces

## Need Help ?

Please, check our dCode Discord community for help requests!
NB: for encrypted messages, test our automatic cipher identifier!