Similar to matrix-vector multiplication, there are multiple ways to interpret matrix-matrix multiplication. The most common one (and by definition) is the inner product interpretation:

Interpret Matrix Multiplication as a Set of Matrix-vector Multiplications

For matrix vector multiplication , if we view as columns, we can view the columns of as matrix-vector products between and the columns of

Analogously, if we represent by rows, we can view the rows of as the matrix-vector product between the rows of and