# Sufficient conditions for symmetry of arbitrary product of real, symmetric, positive semidefinite matrices

It is straightforward to prove that the product $A_1 A_2\cdots A_k$ of $k$ (different) symmetric, real, positive semidefinite matrices is also symmetric if $A_i A_j=A_j A_i$ for all $i,j$. Moreover, it is well-known that for the case $k=2$, this pairwise commutativity condition is also a sufficient condition for symmetry of the matrix product.

My question is the following: Is there a result for $k>2$ concerning sufficient conditions for the symmetry of the product $A_1 A_2\cdots A_k$ of $k$ symmetric, real, positive semidefinite matrices? I have a set of $k$ matrices whose product I know is symmetric, but I would like to know if there's a result in the literature placing any restrictions on the individual matrices $A_i$. I suspect it has to do with pairwise commutativity, but have not been able to figure it out.

Thanks in advance for any insights!

For your first paragraph, the result is not restricted to $k=2$. If each $A_i$ commutes with each $A_j$ then it makes sense to write the product as $$\prod_nA_n$$ as the order does not matter. Since each $A_n$ is symmetyric we have $$\left(\prod_nA_n\right)^\top=\prod_nA_n^\top=\prod_nA_n$$ Thus this also gives an answer to your question, namely that pairwise commutativity is a sufficient condition with symmetric matrices. Now if you start with the knowledge that the product is symmetric, this would not imply each commutes with the other.