What is the correct definition of the standard inner product of two given matrices?
$\endgroup$2 Answers
$\begingroup$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.
$\endgroup$ 1 $\begingroup$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.
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