Matrix Product#

Definition: Matrix Product

The matrix product \(AB\) of a matrix \(A \in F^{m \times n}\) with a matrix \(B \in F^{n \times p}\) is another matrix \(C \in F^{m \times p}\) defined as follows:

\[\begin{bmatrix}a_{11} & \cdots & a_{1n} \\ \vdots & \ddots & \vdots \\ a_{m1} & \cdots & a_{mn} \end{bmatrix}\begin{bmatrix}b_{11} & \cdots & b_{1p} \\ \vdots & \ddots & \vdots \\ b_{n1} & \cdots & b_{np} \end{bmatrix}\overset{\text{def}}{=} \begin{bmatrix} c_{11} & \cdots & c_{1p} \\ \vdots & \ddots & \vdots \\ c_{m1} & \cdots & c_{mp}\end{bmatrix},\]

where

\[c_{ij} \overset{\text{def}}{=} \sum_{k=1}^n a_{ik}b_{kj}\]

Notation

\[C = A\times B = A\cdot B = AB\]

Tip: Computation of Individual Entries

The entry \(c_{ij}\) in the \(i\)-th row and the \(j\)-th column of the matrix product \(C\) is obtained by multiplying the \(k\)-th entry in the \(i\)-th row of \(A\) with the \(k\)-th entry in the \(j\)-th column of \(B\) and summing these products.

Tip: Requirements for Matrix Multiplication

In order for the matrix product \(AB\) to be possible, the number of columns in \(A\) must be equal to the number of rows in \(B\).

Tip: Rows and Columns in the Matrix Product

The matrix product \(AB\) has the same number of rows as \(A\) and the same number of columns as \(B\).

Theorem: Non-Commutativity of the Matrix Product

In general, the matrix product of two matrices is not commutative:

\[A\cdot B \ne B \cdot A\]

Proof

TODO

Theorem: Associativity of the Matrix Product

The matrix product is associative:

\[A \cdot (B \cdot C) = (A \cdot B) \cdot C\]

Proof

TODO

Theorem: Distributivity of the Matrix Product

The matrix product is distributive over matrix addition:

\[\begin{aligned}(A + B)\cdot C &= A\cdot C + B \cdot C \\ A\cdot (B + C) & = A\cdot B + A \cdot C\end{aligned}\]

Proof

TODO