Equivalences

Suppose $Q$ and $X$ are unitary matrices, i.e., $Q^{-1} = Q^*$ and $X^{-1} = X^*$. If $Q$ and $X$ are real, then $Q^{-1} = Q^T$ and $X^{-1} = X^T$, and we call them orthogonal matrices. Let $B = QAX^*$. We say that $B$ is unitarily (orthogonally) equivalent to $A$ and that $Q$ and $X$ are unitary (orthogonal) equivalence transformations. $B$ has the same singular values as $A$. If $u$ and $v$ are left and right singular vectors of $A$, respectively, so that $Av = \sigma u$ and $A^* u = \sigma v$, then $Qu$ and $Xv$ are left and right singular vectors of $B$, respectively.