A matrix decomposition problem similar to the CS decomposition

Question

For two column orthonormal $X,Y\in\mathbb{C}^{n\times k},k<n$, prove that there exist unitary $Q\in\mathbb{C}^{n\times n},U,V\in\mathbb{C}^{k\times k}$, such that $QXU= \begin{bmatrix} I_k\\ 0 \end{bmatrix}$, and $QYV=\begin{bmatrix} \Gamma\\ \Sigma\\ 0 \end{bmatrix}$, $\Gamma,\Sigma\in\mathbb{R}^{k\times k}$ if $k\le n/2$, or $QYV=\begin{bmatrix} \Gamma & 0\\ 0 & I_{2k-n}\\ \Sigma & 0 \end{bmatrix}$, $\Gamma,\Sigma\in\mathbb{R}^{(n-k)\times (n-k)}$ if $k> n/2$. Here $\Sigma,\Gamma$ are nonnegative diagonal and $\Gamma^2+\Sigma^2=I$.

I tried to use CS decomposition for this problem, however it don't work and I don't think CS decomposition can be directly used here since CS decomposition needs $k\le n/2$. I think some ideas of CS decomposition might be useful but I don't know what to use. Can anyone give this problem a hint or something? Thanks!

Answer 1

You can use the following consequence of the CS decomposition.

Let $W$ be an $n \times n$ unitary matrix partitioned as $$ W = \pmatrix{W_{11} & W_{12}\\ W_{21} & W_{22}}, $$ where $W_{11}$ has size $\ell \times \ell$ with $\ell \geq n/2$. Then there exist unitary matrices $U = \operatorname{diag}(U_1,U_2)$ and $V = \operatorname{diag}(V_1,V_2)$ (where $U_1,V_1$ are $\ell \times \ell$) such that $$ U^*WV = \pmatrix{C & 0 & -S\\0 & I_{2\ell - n} & 0\\S & 0 & C}, $$ where $C,S$ are of size $(n-\ell) \times (n-\ell)$ and $C,S$ are diagonal with non-negative entries such that $C^2 + S^2 = I$.