I’m reading wikipedia's Conjugate gradient method page.
On that page, they define an inner product.
$\left<\mathbf{u}, \mathbf{v}\right>_\mathbf{A} := \left<\mathbf{Au}, \mathbf{v}\right> $
And claimes
$\left<\mathbf{Au}, \mathbf{v}\right> = \left<\mathbf{u}, \mathbf{A^Tv}\right> $
($\mathbf{A}$ is symmetric and positive definite)
I cannot understand this process. Why is this true?
Update
I think I solved.
$ \left<\mathbf{Au}, \mathbf{v}\right> = \overline{\overline{ \left<\mathbf{Au}, \mathbf{v}\right> }} = \overline{\overline{ \mathbf{A}\left<\mathbf{u}, \mathbf{v}\right> }} = \overline{\mathbf{A^T}\left<\mathbf{v},\mathbf{u}\right> } = \overline{\left<\mathbf{A^Tv}, \mathbf{u}\right>} = \left<\mathbf{u}, \mathbf{A^Tv}\right> $
I hope this is right.