I'm trying to find an example to show that the product of two non-zero symmetric matrices can be anti-symmetric.
I've proven that this is impossible for 2x2 matrices.
For 3x3 matrices, I've formulated a set of linear equations in 12 variables and used MATLAB to try and find a solution, to no avail.
So, is this possible, and if so, what is the best method to use to formulate an example? If not, what is the best way to prove that it is impossible for matrices of size n (n arbitrary natural number)?
With very many thanks,