Equivalences (Similarities)

Suppose $Q$ is a unitary matrix, i.e., $Q^{-1} = Q^*$. If $Q$ is real then $Q^{-1} = Q^T$ and we say that $Q$ is an orthogonal matrix. Let $B = Q^*AQ$. We say that $B$ is unitarily (orthogonally) similar to $A$, and that $Q$ is a unitary (orthogonal) similarity transformation. If $A$ is Hermitian, so is $B$, and it has the same eigenvalues. The similarity transformation corresponds to introducing a new basis with the columns of $Q$ as vectors. If $y$ is an eigenvector of the transformed matrix $B$, then $x=Qy$ is an eigenvector of the original matrix $A$.