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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?


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A necessary and sufficient condition is that they commute. – Did Mar 30 '12 at 16:08
By the way, for your future reference, this is a much better place to ask this kind of question than mathoverflow. – Neal Mar 30 '12 at 16:25
up vote 2 down vote accepted

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 $\forall X \in \mathbb{R}^n, <X,AX> = <A^*X,X>$, where < _ , _ > 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.

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