So in ${\bf R}^{n\times p}$ we have the Frobenius inner product given by $$\langle A, B\rangle=\text{tr}(A^TB)$$
which can be interpreted as the Euclidean inner product on ${\bf R}^{np}$. My understanding is that all inner products on ${\bf R}^{np}$ can be written as $$a^TPb$$ for $P$ positive-definite. The best I could do in attempting to extend the Frobenius inner product on ${\bf R}^{n\times p}$ is something of the form $$\langle A, B\rangle=\sum_{i=1}^N\text{tr}((X_iAY_i)^T(X_iBY_i))$$ for $X_i\in{\bf R}^{m_i\times n}$ and $Y_i\in{\bf R}^{p\times q_i}$ all full rank. However I would like to know if this covers all inner products on ${\bf R}^{np}$, or if maybe it's more complex than necessary due to redundancies.
I can find the corresponding $P$ matrix for any specific matrix inner product by taking the standard basis for ${\bf R}^{n\times p}$ and forming the matrix
\begin{bmatrix} \langle E_1,E_1\rangle & \langle E_1,E_2\rangle & \dots & \langle E_1,E_{np}\rangle \\ \langle E_2,E_1\rangle & \langle E_2,E_2\rangle & & \vdots \\ \vdots & & \ddots \\ \langle E_{np},E_1\rangle & \dots & \dots & \langle E_{np},E_{np}\rangle \end{bmatrix}
but I don't know if the general form for a matrix inner product I gave above covers all positive-definite matrices $P$.
Update:
newer version of this question on MathOverflow: https://mathoverflow.net/questions/229675/extending-the-trace-inner-product-to-all-matrix-real-inner-products