Dr. Gilbert Strang

The wonderful thing about matrix multiplication is you can do it alot of ways, it comes out the same every way, and each way tells you something.

Mental Model

If $A$ is a matrix, what do u think it is?

A collection of numbers arranged in a grid?

And then multiplying it by another matrix $B$ is some operation that spits out another matrix $C$ that is another grid of numbers?

And numbers in $C$ are computed by moving through $A$ and $B$ in some defined way that seems arbitrary and can be easily forgotten?

This is a totally valid view to have. However, a little shift in perspective may give u a more intuitive understanding of the multiplication operation and will reveal it as as not a single operation, but 4 different types of operations.

What a Matrix is

Bringing back our matrix $A$, it can be viewed as a collection of either columns or rows.

Multiplying it by another matrix $B$ that’s either a Rows or Columns matrix results in 4 different types of operations (depending on how each matrix is viewed)

Matrix Multiplication as Inner Products

$A\ast B=$ Rows matrix * Columns matrix $=\left[\begin{array}{ccc}\leftarrow & {\vec{a}}_0& \to \\ \leftarrow & {\vec{a}}_1& \to \end{array}\right]\ast \left[\begin{array}{cc}\uparrow & \uparrow \\ {\vec{b}}_0& {\vec{b}}_1\\ \downarrow & \downarrow \end{array}\right]=C$

$C$ contains the inner product of every vector in $A$ with each vector in $B$.

Matrix Multiplications as Outer Products

$A\ast B=$ Columns matrix * Rows matrix $=\left[\begin{array}{cc}\uparrow & \uparrow \\ {\vec{a}}_0& {\vec{a}}_1\\ \downarrow & \downarrow \end{array}\right]\ast \left[\begin{array}{ccc}\leftarrow & {\vec{b}}_0& \to \\ \leftarrow & {\vec{b}}_1& \to \end{array}\right]=C$

$C$ is the sum of the outer product matrix of every vector in $A$ with each vector in $B$.

Matrix Multiplication as Column Combinations

$A\ast B=$ Columns matrix * Columns matrix $=\left[\begin{array}{cc}\uparrow & \uparrow \\ {\vec{a}}_0& {\vec{a}}_1\\ \downarrow & \downarrow \end{array}\right]\ast \left[\begin{array}{cc}\uparrow & \uparrow \\ {\vec{b}}_0& {\vec{b}}_1\\ \downarrow & \downarrow \end{array}\right]=\left[\begin{array}{cc}\uparrow & \uparrow \\ A\ast {\vec{b}}_0& A\ast {\vec{b}}_1\\ \downarrow & \downarrow \end{array}\right]=C=\left[\begin{array}{cc}\uparrow & \uparrow \\ {\vec{c}}_0& {\vec{c}}_1\\ \downarrow & \downarrow \end{array}\right]$

$A$ multiplies every column vector in $B$.

Matrix Multiplication as Row Combinations

$A\ast B=$ Rows matrix * Rows matrix $=\left[\begin{array}{ccc}\leftarrow & {\vec{a}}_0& \to \\ \leftarrow & {\vec{a}}_1& \to \end{array}\right]\ast \left[\begin{array}{ccc}\leftarrow & {\vec{b}}_0& \to \\ \leftarrow & {\vec{b}}_1& \to \end{array}\right]=\left[\begin{array}{ccc}\leftarrow & {\vec{a}}_0\ast B& \to \\ \leftarrow & {\vec{a}}_1\ast B& \to \end{array}\right]=C=\left[\begin{array}{ccc}\leftarrow & {\vec{c}}_0& \to \\ \leftarrow & {\vec{c}}_1& \to \end{array}\right]$

Every row vector in $A$ multiplies $B$.

Connections