Geometrical or Physical significance (interpretation) of the inner-product $\langle A,B \rangle := Trace (AB^t)$ over $M_n(\mathbb R)$

$\langle A,B \rangle := Trace (AB^t)$ is an inner product over the vector space $M_n(\mathbb R)$ of all real matrices of size $n$ , I would like to know whether this inner-product has any Geometrical or Physical significance (interpretation) or not ? Please shed some light . Thanks in advance

Solution 1:

For this inner product, the squared norm of a matrix will be the sum of all squared coefficients of the matrix, so this norm, and the inner product, correspond to the canonical inner product and $2$-norm on $\mathbf{R}^{n^2}$, when the latter is identified with the $M_n (\mathbf{R})$. This is the simple euclidian norm !