I need to find a way to get a non negative matrix from an arbitrary matrix only by matrix multiplication, something like: $Y$ is an arbitrary matrix, find matrices $M$(and $N$) $\neq 0 $ so that entries of matrix $ Q= M.Y$ or $Q =M.Y.N$ nonnegative

  • 4
    $\begingroup$ set M = 0 then you'll always get a nonnegative matrix $\endgroup$
    – Surb
    Sep 16, 2015 at 8:35
  • $\begingroup$ By "nonnegative", do you mean "entrywise nonnegative" or "nonnegative definite" (i.e. positive semidefinite)? Anyway, in either case, as Surb points out, set $M=0$ and you get a matrix that is both entrywise nonnegative and nonnegative definite. $\endgroup$
    – user1551
    Sep 16, 2015 at 9:37
  • $\begingroup$ i mean entrywise nonnegative $\endgroup$
    – vegeta
    Sep 16, 2015 at 10:28

1 Answer 1


This can always be done, e.g. by using singular value decomposition.

More generally, for any real or complex matrix $Y$, there always exist two invertible matrices $P$ and $Q$ such that $Y=PDQ$, where $D$ is a (possibly rectangular) diagonal matrix whose diagonal entries belong to $\{0,1\}$. Hence $P^{-1}YQ^{-1}$ is $D$, which is entrywise nonnegative.


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