Matrix Decomposition Question: If $A$ is symmetric when is it true that $A = B^T B $

Question

I was thinking about the fact that for a symmetric matrix A the maximum values of $X^T A X$, where $\|X\| = 1$ occurs when $X$ is an eigenvalue of $A$. I was trying to think of a good geometric image of why this is the case.

One thing that I think might help is in the special case that $A = B^T B$ for some matrix $B$ then we have that $f(X) = \|BX\|$. Then, if one pictures the image of the unit sphere under $B$, it is an ellipsoid and the maxima of $f$ correspond to major axes of the ellipsoid.

So, just wondering, under what circumstances is it possible to write a symmetric matrix $A$ as $A = B^T B$ for some $B$?

Answer 1

That's exactly when $A$ is positive semi-definite. See Cholesky decomposition.