Suppose $X$ is $p_1\times p_2$ matrix with rank $r$. Consider the block matrix $X=\begin{pmatrix}X_{11} & X_{12}\\ X_{21} & X_{22}\end{pmatrix}$. Suppose $X_{11}$ is $k_1\times k_2$ matrix with rank $r$ ($r\leq k_1,k_2$). Prove that the representation $X_{22}=X_{21}X_{11}^\dagger X_{12}$ is unique, where $^\dagger$ means the Moore-Penrose inverse of a matrix.
My current idea is the following: if $X$ and $X_{11}$ is a square matrix, then $X_{22}=X_{21}X_{11}^{-1} X_{12}$ and is unique. However, the statement I'd like to prove is a more generalized version of the above. I am also not sure how to show uniqueness.
I'll keep posted if I have progress. Any help will be appreciated.
It isn't clear what the uniqueness question here refers to, because every matrix has a unique Moore-Penrose pseudo-inverse. Anyway, since $$ \operatorname{rank}(X)\ge\operatorname{rank}\pmatrix{X_{11}&X_{12}}\ge\operatorname{rank}(X_{11})=\operatorname{rank}(X), $$ we must have $$ \operatorname{rank}(X)=\operatorname{rank}\pmatrix{X_{11}&X_{12}}=\operatorname{rank}(X_{11}).\tag{1} $$ It follows from the first equality on $(1)$ implies that the rows of $\pmatrix{X_{21}&X_{22}}$ lie inside the row space of $\pmatrix{X_{11}&X_{12}}$, i.e., $\pmatrix{X_{11}&X_{12}}=Z\pmatrix{X_{11}&X_{12}}$ for some matrix $Z$. Similarly, the second equality on $(1)$ implies that the columns of $X_{12}$ lie inside the column space of $X_{11}$. Therefore $X_{12}=X_{11}Y$ for some matrix $Y$. Consequently, $$ X=\pmatrix{X_{11}&X_{11}Y\\ ZX_{11}&ZX_{11}Y} $$ and $$ X_{22}=ZX_{11}Y=Z(X_{11}X_{11}^+X_{11})Y=(ZX_{11})X_{11}^+(X_{11}Y)=X_{21}X_{11}^+X_{12}. $$