what are the conditions for the product of 2 symmetric matrices being symmetric 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!
 A: The problem can be conveniently approached using the adjoint properties of symmetric and Hermitian matrices.  Over a real vector space, if $A$ is a symmetric matrix, it equals its adjoint, $A^*$.  Remember, the definition of the adjoint matrix $A^*$ is that it is the unique matrix such that $\forall X \in \mathbb{R}^n, \langle X,AX\rangle = \langle A^*X,X\rangle$, where $\langle \_ , \_\rangle$ denotes the inner product in Euclidean space that induces the standard metric.
Directly from the definition, we can show that the adjoint of a product of two matrices is the product of the adjoints in reverse order, or $(AB)^* = B^*A^*$. and if $A$ and $B$ are symmetric matrices, $(AB)^* = B^*A^* = BA$.  If their product, $AB$, is symmetric, then $(AB)^* = AB$, so for this to occur, $AB$ must equal $BA$, or the matrices must commute.
Now, a matrix $A$ is Hermitian if $A^* = A$ with respect to the inner product over $\mathbb{C}^n$, and so the same result is true of products of two Hermitian matrices;   their product is Hermitian iff the matrices commute.
