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 .