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> [!proposition] Proposition. ([[singular values are unitarily invariant]])
> Let $A \in \mathbb{F}^{M \times M}$. Suppose $W \in \mathbb{F}^{M \times M}$ and $Q \in \mathbb{F}^{N \times N}$ are each [[isometric matrix|unitary]] [[matrix|matrices]]. Then $A$ and $C:=WAQ$ have the same [[singular values]].
> [!proof]- Proof. ([[singular values are unitarily invariant]])
Using the [[Singular Value Decomposition of a Matrix|SVD]], write $A=U\Sigma V'$. Then $\begin{align}
C = & WAQ \\
= & (WU)\Sigma (VQ) \\
= & U^{(1)} \Sigma V^{(1)'} \ (1)
\end{align}$
where $U^{(1)}:=WU$ and $V^{(1)}=VQ$. These two matrices are products of [[orthonormal|]] unitary [[matrix|matrices]] and are hence unitary. Thus the expression $(1)$ is an [[Singular Value Decomposition of a Matrix|SVD]] of $C$, from which we see that $C