My intuition tells me that the zero matrix is the only matrix that is symmetric and nilpotent with real values, but I'm having trouble proving it (or finding a counterexample.)
I have searched for related problems, but I've found only one where nilpotent was defined as any matrix $A$ where $A^2=0$; using this definition, the problem is pretty easy. I'm using the more general definition that $A$ is nilpotent if and only if there exists a positive integer $k$ such that $A^k=0$.
Based on my observations while trying to find a counterexample, I've been trying to formulate some argument about the positive semi-definiteness of the entries on the main diagonal, but I'm not getting very far with it. Is this the right approach? Is my gut feeling even true?
Best Answer
Hint: Real symmetric matrices are (orthogonally) diagonalisable. And all eigenvalues of nilpotent matrices are zero.