Matrix Multiplication

$A_{m \times n} \cdot B_{n \times p}$ :

Precondition: # columns of A equals # rows of B.
Algorithm: $(A \cdot B)[i,j] = A[i,1] B[1,j] + A[i,2] B[2,j] + \cdots + A[i,n] B[n,j]$
Alternatively, write A as an array of row vectors $\begin{eqnarraystar} A & = & \left[ \begin{array} {c} a_1^T \ a_2^T \ \vdots \ a_m^T \end{array} \right] \end{eqnarraystar}$ and B as an array of column vectors $\begin{eqnarraystar} B & = & \left[ b_1 \; b_2 \; \cdots \; b_p \right], \end{eqnarraystar}$ then $\begin{eqnarraystar} A \cdot B & = & \left[ \begin{array} {cccc} a_1^T b_1 & a... ...T b_1 & a_m^T b_2 & \cdots & a_m^T b_p \end{array} \right]. \end{eqnarraystar}$
# operations :
- $A \cdot B$ involves $(m \times p)$ dot products, and
- Each dot product needs O(n) operations.

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