Standard inner product of matrices

Question

What is the correct definition of the standard inner product of two given matrices?

Answer 1

According to wikipedia the standard matrix inner product on square matrices is defined as $\langle A,B\rangle=tr(AB^t)$. The properties are also proved here.

Answer 2

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.