The definition of an eigenvector of a square matrix is a vector satisfying the relation
We can rearrange this equation to obtain
The zero vector trivially satisfies this equation for all . Asserting that the trivial solution is not an eigenvector is equivalent to saying that the set of eigenvectors of is the set of vectors satisfying this relation when is singular, i.e.,
Since the determinant of an -dimensional square matrix of size can be expressed as an -degree polynomial with up to distinct roots, we know that there must be at most distinct eigenvalues for . These can either be obtained either analytically or (more typically) numerically.
For each eigenvalue so obtained, we can obtain its corresponding eigenvector by solving
Since this relation holds for any scaling factor , we can take the eigenvectors to all be of unit length without loss of generality.