Outer form of matrix-matrix product

Product of matrices as a sum: $A=BC$. We can rewrite this product in the following form:

$$a_{ij}=\sum _k{b}_{ik}{c}_{kj}=b_{i1}{c}_{1j}+b_{i2}{c}_{2j}+\dots$$

• $b_{,k}$: column $k$ of $B$.
• $c_{k,}$: row $k$ of $C$.

The expression above can be rewritten as:

$$A=\sum _k{b}_{,k}{c}_{k,}$$

This is the outer form of the product. This will be useful for several matrix factorizations, including the LU and QR factorizations.

Matrix-vector and matrix-matrix product