Consider the SVD of matrix $A$:
$$A = U \Sigma V^\top$$
If $A$ is a symmetric, real matrix, is there a guarantee that $U = V$?
There is a similar question here that also posits $A$ is positive semi-definite. But I'm wondering whether $U$ would be equal to $V$ if $A$ is symmetric?
Best Answer
No: if $\Sigma$ is diagonal with non-negative entries, then $U \Sigma U^T$ will necessarily be positive semidefinite. Indeed, we note that for any column vector $x$, we have $$ x^\top(U\Sigma U^\top)x = (U^\top x)^\top \Sigma (U^\top x) \geq 0. $$