A matrix defines a self adjoint operator if and only if it is symmetric

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I know this is probably a silly question, but I'm stuck with it.

$\forall x,y \in \mathbb{R}^{n}$, and $A$ real matrix ($n\times n$), then $\langle Ax,y\rangle=\langle x,Ay\rangle \iff A^{T}=A$

I've tried to prove it using coordinates, but somehow I haven't managed to get the result. Also, I'm sure there must be a shorter way to prove this result.

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2 Answers

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Hint: What is $<A e_i, e_j>$? What is $<e_i, Ae_j>$? (Recalling that the definition of $A$ is $(a_{ij})$ with the property that $A e_i = \Sigma_{j = 1}^{n} a_{ij} e_j$ (or whatever order you put the indices in)).

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Note $\langle Ax,y\rangle=(Ax)^Ty=x^TA^Ty$ and $\langle x,Ay\rangle=x^TAy$. The result follows from these two observations.

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