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

Eigenspaces of a Matrix - dCode

Tag(s) : Matrix

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*!

⮞ Go to: Eigenvalues of a Matrix

⮞ Go to: Eigenvectors of a Matrix

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 $.

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\} $

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.

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.

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 2024-10-05,

eigenspace,eigen,space,matrix,eigenvalue,value,eigenvector,vector

https://www.dcode.fr/matrix-eigenspaces

© 2024 dCode — El 'kit de herramientas' definitivo para resolver todos los juegos/acertijos/geocaching/CTF.

Feedback