Orthogonal invariance of the Euclidean norm
= Orthogonal invariance of the Euclidean norm
{title2=$\|Qx\|_2=\|x\|_2$}
For an <orthogonal matrix> $Q$, the identity $Q^TQ=I$ gives $\|Qx\|_2^2=x^TQ^TQx=x^Tx$. Thus multiplication by $Q$ preserves the <Euclidean norm>. Applying this on both input and output sides proves $\|UAV^T\|_2=\|A\|_2$ for <orthogonal matrices> $U,V$ and the Euclidean <operator norm>. The analogous invariance of the <Frobenius norm> follows by summing column lengths or by a <trace> calculation.