Symmetric matrix and inner product: $\langle Ah,x\rangle = \langle h,A^T x\rangle =\langle Ax,h\rangle$

Question

If A is real, symmetric, regular, positive definite matrix in $R^{n.n}$ and $x,h\in R^n$, why is it $\langle Ah,x\rangle = \langle h,A^T x\rangle =\langle Ax,h\rangle$? Is there some rule or theorem for this?

Answer 1

Note that inner product can be written as: $\langle x,y\rangle=y^Tx$. So $\langle Ah, x\rangle=x^T Ah$ and $\langle h, A^T x\rangle=(A^Tx)^Th=(x^TA)h=\langle Ah,x\rangle.$ Also $A^T=A$ as $A$ is symmetric and this gives the last equality.

Answer 2

I think this follows straightforward from definitions:

$\,A\,$ is symmetric and real$\;\implies A^*=A^t=A\;$ , so since the inner product is real we get

$$\langle x,y\rangle =\langle y,x\rangle \;\;\text{ and}\;\;\langle y,A^tx\rangle=\langle A^tx,y\rangle=\langle Ax,y\rangle\;\ldots $$