# Eigenvalues and eigenvectors

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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.