Let $A \in \mathbb{C}^{n \times n}$, $X \in \mathbb{C}^{n\times k}$, $M\in \mathbb{C}^{k\times k}$. If $X^HX=I_k$ and $\|\cdot\|$ is unitarily invariant, i.e., if for any unitary matrices $U, V$ we have $$\|UYV\|=\|Y\| , \quad \forall Y\in\mathbb{C}^{n\times k}$$ prove that
$$X^H A X = \arg \min_M \| AX - XM \|$$
I know that $XX^HAX$ is the projection of $AX$ into $R(X)$, but I have no idea to use the properties of unitarily invariant norm. Any help will be appreciated.
Hint: Note that there exists an $n \times n$ unitary matrix $W$ such that $X = WJ$, where $$ J = \pmatrix{I_k\\0}. $$ Thus, we have $$ X^HAX = (WJ)^HA(WJ) = J^H(W^HAW)J $$ and also $$ \|AX - XM\| = \|A(WJ) - (WJ)M\| \\ = \|W^H(AWJ - WJM)\| = \|(W^HAW)J - JM\|. $$ In other words, it suffices to prove the following: if $B = W^HAW$, then $$ J^H B J = \arg\min_M \|BJ - JM\|. $$ If we divide $B$ into blocks so that $$ B = \pmatrix{B_{11} & B_{12}\\ B_{21} & B_{22}}, \quad B_{11} \in \Bbb C^{k \times k} $$ then we find that $$ J^HBJ = B_{11}, \quad BJ - JM = \pmatrix{B_{11} - M\\B_{21}}. $$