We know for a skew-symmetric matrix A, $x^TAx = 0$. But is the converse statement true, i.e. does $x^TAx = 0$ imply A is skew-symmetric? If yes, then how to prove it?
[Math] Prove $x^TAx = 0$ $\implies$ A is skew-symmetric
linear algebramatricestranspose
Best Answer
It is true. We have: $$(x+y)^TA(x+y) = 0 \implies x^TAx + y^TAx + x^TAy + y^TAy = 0.$$But $x^TAx = y^TAy = 0$, so we have: $$x^TAy = -y^TAx.$$Take $x = e_i$ and $y = e_j$ to get $a_{ij} = -a_{ji}$.