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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.

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  • $\begingroup$ 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 $\endgroup$ Mar 24, 2022 at 23:04
  • $\begingroup$ The singular value decomposition is a prominent example, this decomposition always exists. $\endgroup$ Mar 24, 2022 at 23:05
  • $\begingroup$ Let $A$ and $C$ be any invertible matrices and let $B = A^{-1}MC^{-1}$. This will rarely be useful. $\endgroup$ Mar 24, 2022 at 23:06
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    $\begingroup$ 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 $C$. $\endgroup$
    – Rob Arthan
    Mar 24, 2022 at 23:16
  • $\begingroup$ @RobArthan love the simplicity of this answer $\endgroup$ Mar 24, 2022 at 23:28

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