Matrix Product

Definition: Matrix Product

The matrix product 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:

$$\left[\begin{array}{ccc}{a}_{11}& \cdots & a_{1n}\\ ⋮& \ddots & ⋮\\ a_{m1}& \cdots & a_{mn}\end{array}\right]\left[\begin{array}{ccc}{b}_{11}& \cdots & b_{1p}\\ ⋮& \ddots & ⋮\\ b_{n1}& \cdots & b_{np}\end{array}\right]\stackrel{\text{def}}{=}\left[\begin{array}{ccc}{c}_{11}& \cdots & c_{1p}\\ ⋮& \ddots & ⋮\\ c_{m1}& \cdots & c_{mp}\end{array}\right],$$

where

$$c_{ij}\stackrel{\text{def}}{=}\sum _{k=1}^n{a}_{ik}{b}_{kj}$$

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 - for all matrices $A\in F^{m\times n},B\in K^{n\times p}$ and $C\in K^{p\times q}$

$$A\cdot (B\cdot C)=(A\cdot B)\cdot C$$

PROOF

TODO

Theorem: Distributivity of the Matrix Product

The matrix product is distributive over addition:

$$\begin{array}{rl}(A+B)\cdot C& =A\cdot C+B\cdot C\\ A\cdot (B+C)& =A\cdot B+A\cdot C\end{array}$$

PROOF

TODO