The th eigenvector of a matrix has the defining property
where is the associated eigenvalue. Thus, the eigenvector is invariant under the linear transformation to within a (generally complex) scale factor .
An matrix typically has eigenvectors.1Let's make a similarity-transformation matrix out of the eigenvectors:
Then we have
where diag is a diagonal matrix having along its diagonal. Premultiplying by gives
Thus, is a similarity transformation that diagonalizes the system.