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

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 ) will be scalars such that

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

## Example

Given the matrix

The characteristic polynomial will be

The eigenvalues of will be -1 and 3.