Completing matrix $B$ so ${B=PA}$

Question

I have this problem.

${B=PA}$, where $P$ is a $l\times l$ invertible unknown matrix. $A$,$B$ are two $l \times m$ matrices.

All entries of A are known. Some entries of B are known, but some entries are missing. I want to complete these missed entries.

Theoretically, if $m$ is larger than $l$, we can find them by solving the linear equations. My questions is that are there other novel approaches to find these missed elements?

Thanks.

Answer 1

since you know $\mathbf{P}$, why cant you simply take the product of $\mathbf{P}$ with $\mathbf{A}$ and find the missing values in $\mathbf{B}$ ?. If $\mathbf{P}$ is unknown, best thing is to estimate $\mathbf{P}$ using the known values in B using matrix factorization techniques. Since you want P to be invertible, you have to add extra constraints like $det(\mathbf{P}) \neq 0$