Partial derivative of loss function

Question

How I take the partial derivative with respect to $A_i$, B and $C_j$ from the following loss function: $$ L=A_i^T B C_j $$ where $A_i$ and $C_j$ are vectors.(A, B and C are matrices).

Answer 1

I'm not sure. But according to Eqs.(69) and (70) in "The matrix cookbook", I think that :

$$ \frac{∂(L)}{M}=U_iV_j^T $$ $$ \frac{∂(L)}{U_i}=MV_j $$ $$ \frac{∂(L)}{V_i}=M^TU_j $$ .