Suppose is a nonsingular matrix.
Let
.
We say that
is
similar to
and that
is a similarity transformation.
has the same eigenvalues as
.
If
is an eigenvector of
, so that
, then
is an eigenvector of
.
If is a unitary matrix, i.e.,
,
we say
is unitarily similar to
.
If
is real, we say orthogonal instead of unitary.