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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

$A=\begin{bmatrix}3&0\\-1&-1\end{bmatrix}$

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.