Matrix Multiplication

linear-algebra

Definition

Matrix Multiplication

Let $A=(a_{ij})\in F^{m\times n}$ and $B=(b_{jk})\in F^{n\times q}$ be two matrices. Then their product $A\cdot B$ is the $m\times q$ matrix $C=(c_{ik})$ with entries

$$c_{ik}=\sum _{j=1}^n{a}_{ij}{b}_{jk}=⟨A_{i,\ast },B_{\ast ,k}⟩$$

The entry in row $i$ and column $k$ is obtained by taking the dot product of the $i$-th row of $A$ with the $k$-th column of $B$.

Associativity

Associativity

Matrix multiplication is associative whenever the dimensions match. For

$$A\in F^{m\times n},B\in F^{n\times p},C\in F^{p\times q},$$

we have

$$(AB)C=A(BC).$$

Associativity

Entrywise, for $1\le i\le m$ and $1\le \ell \le q$:

$$\begin{array}{rl}((AB)C{)}_{i\ell }& =\sum _{k=1}^p(AB{)}_{ik}{c}_{k\ell }\\ & =\sum _{k=1}^p\left(\sum _{j=1}^n{a}_{ij}{b}_{jk}\right)c_{k\ell }\\ & =\sum _{j=1}^n{a}_{ij}\left(\sum _{k=1}^p{b}_{jk}{c}_{k\ell }\right)\\ & =(A(BC){)}_{i\ell }.\end{array}$$

Thus products such as $ABC$ are unambiguous, even though matrix multiplication is generally not commutative.