# Which type of factorization is appropriate?

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?

• 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. – Robert Israel Feb 28 at 12:50
• @RobertIsrael But what if you allow $Y$ to be complex? Does a decomposition exist in all cases then? – Parcly Taxel Feb 28 at 12:59
• @RobertIsrael Thanks for your response, but what if Y to be complex? And is there anyway to force a matrix to be positive semidefinite? – user137927 Feb 28 at 13:02
• @RobertIsrael Or any approximation for it. – user137927 Feb 28 at 19:33
• 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. – Robert Israel Feb 28 at 19:42