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

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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.

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    The matrix doesn't need to be square. The transpose guarantees that the expression works for any dimensions of the matrix ($A$ and $B$ of course must be of the same dimensions). Also note that this is only for real matrices; for complex matrices you need to use the Hermitian transpose (i.e. transpose + complex conjugation). – 2017-01-26
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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 :

$$\left=\sum_{1\le i,j\le n}a_{i,j}b_{i,j}$$

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.