Eigenvalues and eigenvectors

An eigenvector of an n×n matrix $A$ is a vector that does not change its direction under a linear transformation; that is, if $v$ is a non-zero vector and $\lambda$ is a scalar (the eigenvalue of $A$),

$A\mathbf{v}=\lambda\mathbf{v}$

Eigenvalues can be real or complex. The product of the eigenvalues is the determinant of the matrix, and the linear span of an eigenvector is called an eigenspace.

Computing eigenvectors and eigenvalues

The eigenvalues (represented by $\lambda$) will be scalars such that

$|\lambda I_n-A|=0$

This equation is known as the characteristic polynomial. The eigenvectors corresponding to the eigenvalue $\lambda$ will be the non-trivial solutions to

$\lambda I_n-A=0$

Example

Given the matrix

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The characteristic polynomial will be

$\begin{vmatrix}\lambda I-\begin{bmatrix}3&0\\-1&-1\end{bmatrix}\end{vmatrix}=\begin{vmatrix}\lambda\begin{bmatrix}1&0\\0&1\end{bmatrix}-\begin{bmatrix}3&0\\-1&-1\end{bmatrix}\end{vmatrix}=0$

$\begin{vmatrix}\lambda-3&0\\1&\lambda+1\end{vmatrix}=(\lambda-3)(\lambda+1)-(0)(1)=(\lambda-3)(\lambda+1)=0$

The eigenvalues of $A$will be -1 and 3.