How is the inner product of a matrix space defined?

Question

I will use a link which will save a lot of writing and possibility of typo's on my part. Page 4 first paragraph the textbook defines the matrix space and the last sentence is :

"This inner product is identical to the dot product on Rmn if an m × n matrix is viewed as an mn × 1 matrix by stacking its columns."

If I use matrix multiplication I come up with a vector not a string of numbers. Does anyone have any ideas?

Here is the link by the way.

https://web.auburn.edu/holmerr/2660/Textbook/innerproduct-print.pdf.

Answer 1

The inner product that is being described is

$\langle A, B \rangle=\mbox{tr}(A^{T}B)=\sum_{i=1}^{m} \sum_{j=1}^{n} A_{i,j}B_{i,j} $.

Answer 2

The inner product defined there $\langle A, B \rangle = \sum_{i=1}^m \sum_{j=1}^n a_{ij} b_{ij}$ is not matrix multiplication. It is as described in the quote (the usual inner product on $\mathbb{R}^{mn}$ if you convert $A$ and $B$ into tall column vectors)