Suppose I have a matrix $A$ where each entry is also a matrix, i.e., $A_{ij}\in R^{n\times n}$. Let us suppose every $A_{ij}$ is positive semi-definite as well. Question is: Do there exists $\{B_i\},\{C_j\}$ such that $A_{ij}=B_i C_j$ for every $i,j\in [n]$? Any reference also is welcome.
A version of this is true if we just had a positive semidefinite matrix, then every entry can be written as $A_{ij}=\langle x_i,x_j\rangle$.