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Eigenspaces of a Matrix

Tool to calculate eigenspaces associated to eigenvalues of a matrix.

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# Eigenspaces of a Matrix

## Eigenvectors Calculator

Tool to calculate eigenspaces associated to eigenvalues of a matrix.

### What is an eigenspace of an eignen 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 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 $\text{Vect} \left\{ \begin{pmatrix} -1 \\ 1 \end{pmatrix} \right\}$

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