A problem about the quadratic form $x^TAx=0$.

Question

$A$ is an $m\times m$ real matrix in $\mathbb{R}^{m\times m}$. If $x^TAx=0\ \forall\ x\in\mathbb{R}^m$, can we conclude that $A=0_{m\times m}$? Why?

Note: $x^T$ is the transpose of $x$. $0_{m\times m}$ is an $m\times m$ matrix and its elements are all zero.

Answer 1

As said in the other answers, it is not true for general square matrices but the statement holds if $A$ is symmetric. In fact, every matrix $A\in\mathbb{R}^{n\times n}$ can be split in the sum of symmetric and skew-symmetric matrix: $A=H+S$, $H^T=H$, $S^T=-S$. Note that $\langle x,Ax\rangle = \langle x, Hx\rangle$, so $\langle x,Ax\rangle=0$ for all $x$ means only that $H=0$ and $A$ can be an arbitrary skew-symmetric matrix.

Answer 2

Note that $x^TAx=\langle x,Ax\rangle$ under the usual inner product. What does it mean that $\langle x,Ax\rangle=0$ for all $x\in\mathbb R^m$?

Answer 3

Take $$A=\begin{bmatrix} 0&1\\-1&0 \end{bmatrix}.$$