I'm trying to prove the following statement, which feels very natural, but I am having very little success.
For matrices $A,B$, we have $||A^\top A + B^\top B||_2 \ge || A^\top A ||_2$.
I feel like this should be true, because $B^\top B$ is positive semidefinite, and on an intuitive level I think adding two positive semidefinite matrices can only add more information to the positive singular values. But I cannot prove this; the difficulty is that the SVD of $A$ and $B$ do not interact well. Is there a proof to this? Or is it even true?