Equivalences (Similarities)

Suppose $S$ is a nonsingular matrix. Let $B = S^{-1}AS$. We say that $B$ is similar to $A$ and that $S$ is a similarity transformation. $B$ has the same eigenvalues as $A$. If $x$ is an eigenvector of $A$, so that $Ax = \lambda x$, then $y=S^{-1}x$ is an eigenvector of $B$.

If $S$ is a unitary matrix, i.e., $S^{-1} = S^*$, we say $B$ is unitarily similar to $A$. If $S$ is real, we say orthogonal instead of unitary.