Tool to calculate eigenspaces associated to eigenvalues of any size matrix (also called vectorial spaces Vect).
Eigenspaces of a Matrix - dCode
Tag(s) : Matrix
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Eigenspaces of a Matrix
Eigenspaces Calculator
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.
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.
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Eigenspaces of a Matrix on dCode.fr [online website], retrieved on 2024-11-21,
