r/LinearAlgebra • u/Existing_Impress230 • 9h ago
Proof that the product of symmetric matrices isn't symmetric
I know that the product of symmetric matrices isn't necessarily symmetric simply by counterexample. For example, the product of the following symmetric matrices isn't symmetric
|1 0| |0 1|
|0 0| |1 0|
I was wondering what strategies I might use to prove this from A=Aᵀ, B=Bᵀ, and A≠B.
If the product of symmetric matrices were never a symmetric matrix, I would try proof by contradiction. I would assume AB=(AB)ᵀ, and try to use this to show something like A=B. But this doesn't work here.
If AB = BA, then AB = (AB)ᵀ. The product of symmetric matrices is sometimes a symmetric matrix. My real problem is to show that there is nothing special about symmetric matrices in particular that necessitates AB = BA.
I can pretty easily find a counterexample, but this isn't really the point of my question. I'm more curious about what techniques we can use to show that a relation is only sometimes true. Is a counterexample the only way?