inner product of matrix multiplication

Question

I've seen a few times this property:

$$ <AB, AB> = <A^tAB, B>$$

Is there any proof for this? Can anyone point me to some more material regarding inner-products of matrices?

Answer 1

Yes, the inner-product of two matrices is no different than that of two vectors, i.e., the inner-product of $\newcommand{\reals}{\mathbb{R}}A, B\in\reals^{m\times n}$ is \begin{equation} <A,B> = \sum_{i=1}^m \sum_{j=1}^n A_{ij} B_{ij}. \end{equation}

Now it is straightforward to prove that \begin{equation} \newcommand{\trace}{\mathbf{Tr}} <A,B> = \trace \left(A^T B\right). \end{equation}

Therefore if we use the fact that $(AB)^T = B^T A^T$, we have \begin{equation} <AB,AB> = \trace \left( (AB)^T AB \right) = \trace \left(A^T AB\right)^TB = <A^T AB, B>, \end{equation} hence the proof!