Suppose that $T : W \rightarrow W$ is a linear operator on a finite dimensional complex inner product space. Suppose that $\langle T(x), x \rangle = 0$ for all $x \in W$. Must $T$ must be zero?
I can only prove all eigenvalues of $T$ are zero.
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Another one: https://math.stackexchange.com/q/1839427/42969. – Martin R Dec 10 '17 at 12:17
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This is an interesting question though: if you can prove that all the eigenvalues are zero, i.e. $T$ is nilpotent, try the typical nilpotent nonzero matrix $T = \begin{pmatrix} 0 & 1 \ 0 & 0 \end{pmatrix}$ and see what happens. – Joppy Dec 10 '17 at 12:18