How to prove that $\langle Av,w\rangle=\langle v,A^Tw\rangle$?

Question

How to prove: $$\langle Av,w\rangle=\langle v,A^Tw\rangle$$ $\langle,\rangle$ represents inner-product, $v,w$ denote vectors and $A$ is a matrix.

Answer 1

This only holds for real valued $A$. In general, the following is true: $$ \langle A v,w \rangle = \langle v,A^{\mathsf H} w\rangle $$ where $\mathsf H$ means the conjugate transpose (adjoint) of $A$. This property uniquely defines $A^{\mathsf H}$ from $A$.

To prove your statement, observe that $\langle u,v \rangle = v^\intercal u$. Hence $$ \langle A v,w \rangle = w^\intercal A v = (A^\intercal w)^\intercal v = \langle v,A^\intercal w \rangle $$

Answer 2

I'm assuming you are talking about $n\times n$ real matrices.

Hint: Write down everything component-wise and use the definition of inner product in $\mathbb{R}^n$.

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Alternatively, in $\mathbb{R}^n$, assuming one uses column vectors, one has $$ \langle x,y\rangle=x^Ty $$ for $x,y\in\mathbb{R}^n$.