Let's say I'm asked to give a counterexample for a claim about matrices, such as
The elementwise product of two positive semi-definite matrices is positive semi-definite.
It's easy enough to do some intelligent guess-and-checking with the help of a computer and find an answer in a few seconds.
But how do I do this in my head when I don't have a computer handy? I can't find a simple way. Either I have to pick the eigenvalues in the diagonal matrix $D$ and then somehow come up with a matrix $P$ to make $A = PDP^{-1}$ positive semi-definite, repeat this for another matrix, and hope that the result isn't PSD, or I have to pick a non-PSD $A$ directly (how?), repeat this for a second matrix, and then hope the result is not PSD.
Either way, I have to go through several matrix multiplications just to verify that the result is/isn't PSD, which is trivial for a computer but something that takes too long on a piece of paper.
Is there a better way?
Are there shortcuts I can take to avoid multiplying matrices in my head or on paper?