Column products
Task number: 4470
Consider matrices \(\boldsymbol A\) of type \(m \times n\) and \(\boldsymbol B\) of type \(n \times p\). Let us denote by \(\boldsymbol a_1,\boldsymbol a_2,\dots, \boldsymbol a_n\) the columns of the matrix \(\boldsymbol A\) and \(\boldsymbol b_1^\mathsf T, \boldsymbol b_2^\mathsf T,\dots, \boldsymbol b_n^\mathsf T\) the rows of the matrix \(\boldsymbol B\). (I.e. \(\boldsymbol b_1,\boldsymbol b_2,\dots, \boldsymbol b_n\) are the columns of \(\boldsymbol B^\mathsf T\).)
Which matrix corresponds to the expression \(\boldsymbol a_1\boldsymbol b_1^\mathsf T+\boldsymbol a_2\boldsymbol b_2^\mathsf T+\cdots+\boldsymbol a_n\boldsymbol b_n^\mathsf T\)?
Here, each of \(\boldsymbol a_i\boldsymbol b_i^\mathsf T\) is a matrix of type \(m \times p\), since we consider column vectors as a matrix of one column.
Solution
Denote \(\boldsymbol C=\boldsymbol a_1\boldsymbol b_1^\mathsf T+\boldsymbol a_2\boldsymbol b_2^\mathsf T+\cdots+\boldsymbol a_n\boldsymbol b_n^\mathsf T\).
The matrix \(\boldsymbol a_k\boldsymbol b_k^\mathsf T\) has on the \(i\)-th row and the \(j\)-th colum the product \(a_{ik}b_{kj}\).
Hence \(c_{ij}=\sum\limits_{k=1}^n a_{ik}b_{kj}\) and consequently \(\boldsymbol C=\boldsymbol{AB}\).
Answer
The given expression corresponds to the product \(\boldsymbol{AB}\).
