Suppose is a unitary matrix, i.e.,
.
If
is real then
and we say that
is an
orthogonal matrix. Let
.
We say that
is
unitarily (orthogonally) similar to
,
and that
is a unitary (orthogonal) similarity transformation.
If
is Hermitian, so is
, and it has the same eigenvalues.
The similarity transformation corresponds to introducing a new basis with
the columns of
as vectors. If
is an eigenvector of the transformed
matrix
, then
is an eigenvector of the original matrix
.