I have a matrix $W$ which can be decomposed as $$ W=\left[ \begin{array}{c|c} U & 0\\ \hline Y&Z \end{array} \right] $$
In which, $U,Z$ are given and have $rank(U)=r_1,rank(Z)=r_2$. How can I design the matrix $Y$, so that $rank(W)$ is maximum? Thank all in advance
This is what I tried Let see the general case first
$$ W=\left[ \begin{array}{c|c} U & X\\ \hline Y&Z \end{array} \right] $$
Assuming that all inverses exist, then
$$W^{-1}=\left[ \begin{array}{} [U-XZ^{-1}Y]^{-1}& U^{-1}X[YU^{-1}X-Z]^{-1}\\ [YU^{-1}-Z]^{-1}YU^{-1} & [Z-YU^{-1}X]^{-1} \end{array} \right] $$
Set $X=0$(zero matrix) then $W^{-1}$ becomes $$W^{-1}=\left[ \begin{array}{} U^{-1}& 0\\ Z^{-1}YU^{-1} & Z^{-1} \end{array} \right] $$
From here, I have no idea to design matrix $W$