# Given matrix $M$, how to find $A$, $B$, $C$ such that $M=ABC$?

My question is actually general: how can one "factorize" a matrix? In particular, given an arbitrary matrix $$M$$, can one write $$M=ABC$$ for some non-zero matrices $$A, B, C$$? If yes, is there some way of finding these matrices ("factors")?

EDIT: After reading some nice comments, I present here the conditions on matrices A, B and C, and these are: $$A = \sqrt{a}$$, $$C=\sqrt{c}$$ and $$B= u v^\dagger$$ where $$u$$ and $$v$$ are some column vectors and $$*$$ denotes the combined operation of complex-conjugation and transposition.

• There are too many to list here. I would refer you to the wikipedia article on matrix decompositions for more information. en.wikipedia.org/wiki/Matrix_decomposition Mar 24, 2022 at 23:04
• The singular value decomposition is a prominent example, this decomposition always exists. Mar 24, 2022 at 23:05
• Let $A$ and $C$ be any invertible matrices and let $B = A^{-1}MC^{-1}$. This will rarely be useful. Mar 24, 2022 at 23:06
• Well, you can take any two of $A$, $B$ and $C$ to be $I$ and the other one to be $M$. The question only becomes interesting if you impose some constraints on $A$, $B$ and $C$. Mar 24, 2022 at 23:16
• @RobArthan love the simplicity of this answer Mar 24, 2022 at 23:28