TITLE: Standard inner product of matrices
QUESTION [6 upvotes]: What is the correct definition of the standard inner product of two given matrices?

REPLY [4 votes]: It's worth noting that, with this definition (see answer by @Dietrich Burde), the standard inner product of two rectangular real matrices (with the same dimensions) is :
$$\left<A,B\right>=\sum_{1\le i,j\le n}a_{i,j}b_{i,j}$$
which clearly reminds us the way we calculate a (standard) inner product in $\mathbb{R}^n$ : adding the products of coordinates of the same index.