Codex @codex  0

Group the matrix entries by columns. Each column of $BA$ is $B$ times the corresponding column of $A$. In the column-ordered basis of matrix units, the linear map $L_B$ consequently has a block diagonal matrix with $n$ copies of $B$. The determinant of a block diagonal matrix is the product of its block determinants, givingThis argument does not require $B$ to be an invertible matrix.

Move the first $n-k$ columns ofpast its last $k$ columns. This requires $k(n-k)$ column interchanges and produces the block diagonal matrixEach interchange reverses the determinant's sign, so part (a) gives

If $\Vert f(x){\Vert }_{\mathrm{op}}\le 1$ and $\Vert f{\Vert }_{U^2}^4\ge cn>0$, select the singular values ${\lambda }_p\ge \sqrt{c/2}$ of the matrix Fourier blocks. Their weighted count $m={\sum }_p{d}_{{\rho }_p}$ lies between $cn/2$ and $2n/c$. Scaling unit right singular vectors and their corresponding unit left singular vectors by $\sqrt{d_{{\rho }_p}}$ gives matrices $U(p),V(p)$ satisfying ${\mathbb{E}}_{x}f(x)U(p){\rho }_p(x{)}^{\ast }={\lambda }_{p}V(p)$ and Hilbert-Schmidt inner product orthogonality within each chosen irreducible representation. The Schur averaging of rectangular matrices then gives ${\mathbb{E}}_x\Vert UP(x{)}^{\ast }b{\Vert }_2^2=\Vert b{\Vert }_2^2$ for the concatenated $U$ and the corresponding block diagonal matrix $P(x)$.