TITLE: what are the conditions for the product of 2 symmetric matrices being symmetric
QUESTION [6 upvotes]: In generally, the product of two symmetric matrices is not symmetric, so I am wondering under what conditions the product is symmetric.
Likewise, over complex space, what are the conditions for the product of 2 Hermitian matrices being Hermitian?
Thanks!

REPLY [4 votes]: Incidentally, every real matrix is the product of two symmetric matrices. (If I remember correctly, I once read about this in a paper by Halmos).