Consider a matrix 
with 
as an
eigenvalue, eigenvector pair. Further, let 
be an arbitrary
non-singular matrix. Define a new vector 
. Then
Therefore, 
is an eigenvalue, eigenvector pair of
and 
is said to be
similar to 
. Such transformations are called
similarity transformations. Similarity transforms preserve the
spectrum and characteristic polynomial of a matrix.
Most methods for calculating the eigenvalues and eigenvectors of a matrix perform a sequence of similarity transformations to convert the original matrix to a simpler form.