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!


1 Answer 1


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.


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