Assume I have matrix
${\bf H}= \begin{bmatrix} {\bf H}_{11} & {\bf H}_{12} \\ {\bf H}_{21} & {\bf H}_{22} \end{bmatrix}$
where ${\bf H_{ij}}$ is a $Nr\times Nt$ matrix $i,j \in[1:2]$. Now assume I have another matrix
${\bf F}= \begin{bmatrix} {\bf f}_{11} & {\bf 0} \\ {\bf 0} & {\bf f}_{22} \end{bmatrix}$
where vector ${\bf f}_{ij}$ is $Nt\times 1$ vector and same for ${\bf 0}$ is $Nt \times 1$.
My goal is to find ${\bf f_{11}}$ and ${\bf f_{22}}$ that can can make the the product
$$ \arg\min_{f_{11},f_{22}} ||{\bf H} {\bf F} - \begin{bmatrix} {\bf H}_{11}{\bf f}_{11} & {\bf 0} \\ {\bf 0} & {\bf H}_{22}{\bf f}_{22} \end{bmatrix}|| $$
Is that possible?
(I am lazy and won't write the matrices in bold.)
Hints:
$HF - \begin{bmatrix} H_{11} f_{11} & 0 \\ 0 & H_{22} f_{22} \end{bmatrix} = \begin{bmatrix} 0 & H_{12} f_{22} \\ H_{21} f_{11} & 0 \end{bmatrix}$
$ \left\|\begin{bmatrix} 0 & X \\ Y & 0 \end{bmatrix} \right\|_F^2 = \|X\|_F^2 + \|Y\|_F^2 $
$\min \| A x \| = 0$