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
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

Eigenspaces Calculator

Loading...
(if this message do not disappear, try to refresh this page)

Eigenvalues Calculator

Eigenvectors Calculator

Answers to Questions (FAQ)

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 and the zero vector. 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\} $

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, 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 as long as you cite dCode!
Cite as source (bibliography):
Eigenspaces of a Matrix on dCode.fr [online website], retrieved on 2023-02-08, 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!

Questions / Comments

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


https://www.dcode.fr/matrix-eigenspaces
© 2023 dCode — The ultimate 'toolkit' to solve every games / riddles / geocaching / CTF.
 
Feedback