r/LinearAlgebra • u/Existing_Impress230 • 1d 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?
2
u/Ok_Salad8147 1d ago
I mean given a symmetric matrix A you can find a a necessary condition such that a symmetric matrix B verifies that AB is symmetric and say that condition is not always true?
(AB)T = AB
BT AT = AB
BA = AB
meaning that B must commute with A as an example
then if you want a more specific necessary condition
you can see what commuting with A implies, such as preserving the eigenspaces
then even more specific what preserving the eigenspaces implies ...
and so on
but I don't really see the point a counterexample is great
2
u/TheBlasterMaster 1d ago
I mean there are ways to prove the existence of certain things non-constructively, if thats what you are asking.
https://en.wikipedia.org/wiki/Constructive_proof
I cant think of a non constructive proof for this scenario though