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I have a symmetric matrix (square)($A$) with positive values and zero on its main diagonal. I need to find a matrix $Y$ which is:

$Y^TY = A$

I don't have any non-negativity constraint on the elements of $Y$. I know just a little about matrix factorization and most of the search results are about non-negative matrix factorization. It should be noted that I can't prove that $A$ is positive semi-definite or something like that to use eigen-vectors. Is there any keyword or solution which help me with the problem?

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  • $\begingroup$ If $Y$ is a real matrix, then $Y^T Y$ is positive semidefinite. So if $A$ is not positive semidefinite, you're out of luck. $\endgroup$ – Robert Israel Feb 28 at 12:50
  • $\begingroup$ @RobertIsrael But what if you allow $Y$ to be complex? Does a decomposition exist in all cases then? $\endgroup$ – Parcly Taxel Feb 28 at 12:59
  • $\begingroup$ @RobertIsrael Thanks for your response, but what if Y to be complex? And is there anyway to force a matrix to be positive semidefinite? $\endgroup$ – user137927 Feb 28 at 13:02
  • $\begingroup$ @RobertIsrael Or any approximation for it. $\endgroup$ – user137927 Feb 28 at 19:33
  • $\begingroup$ If $A$ is a real symmetric matrix, then it has a (complex) symmetric square root $Y$ which can be defined using the holomorphic functional calculus. $\endgroup$ – Robert Israel Feb 28 at 19:42

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