 Orthogonal invariance of the Euclidean norm

For an orthogonal matrix $Q$, the identity $Q^{T}Q=I$ gives $\Vert Qx{\Vert }_2^2=x^T{Q}^{T}Qx=x^{T}x$. Thus multiplication by $Q$ preserves the Euclidean norm. Applying this on both input and output sides proves $\Vert UA{V}^T{\Vert }_2=\Vert A{\Vert }_2$ 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.