Let $U$ be a unitary matrix. Does it mean that for every invertible matrix $A$, $$\Vert AUx\Vert \le \Vert Ax\Vert$$ for every vector $x$?
2026-03-29 15:16:50.1774797410
norm of $AUx$ for unitary $U$
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No. Suppose that we're working on $\mathbb{C}^2$ and that$$U=\begin{pmatrix}0&1\\-1&0\end{pmatrix}\text{ and }A=\begin{pmatrix}0&2\\1&0\end{pmatrix}.$$Then, if $x=(1,0)$, $\|Ax\|=1$, whereas $\|AUx\|=2$.