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 2×2 matrices.
For 3×3 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,
Froskoy.
Best Answer
Of course, it's not possible in dimension $1$. If the dimension $d$ is greater than $2$, then let $A=(a_{l, r})_{1\leq l,r\leq d}$ and $B=(b_{l, r})_{1\leq l,r\leq d}$, with $a_{1,1}=1$, $b_{2,2}= 1$ and all the other entries are $0$. Then $A$ and $B$ are non-zero, symmetric and $AB=0$, which is skew-symmetric.