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##

Equivalences (Similarities)

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 .

Susan Blackford
2000-11-20