Matrix Multiplications as Outer Products

What is the problem

You have a matrix $A$ that u view as a columns matrix, and want to multiply it by a rows matrix $B^T$.

$$A=\left[\begin{array}{cc}\uparrow & \uparrow \\ {\vec{a}}_0& {\vec{a}}_1\\ \downarrow & \downarrow \end{array}\right]B^T=\left[\begin{array}{ccc}\leftarrow & {\vec{b}}_0& \to \\ \leftarrow & {\vec{b}}_1& \to \end{array}\right]$$

You want to Compute $A\ast B^T$.

Strategy

1. highlight each column vector in $A$ by a line that also shows its index.
   
2. highlight each row vector in $B^T$ by a line that also shows its index.
   

Their multiplication matrix is then constructed by computing the outer product at every intersection.

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Connections

• Vectors Outer Product
• Transposing a Matrix
• Matrix Multiplication as Inner Products