Matrix Multiplication

The product ${\hbox{\sf C}}$ of two Matrices ${\hbox{\sf A}}$ and ${\hbox{\sf B}}$ is defined by

$$c_{ik} = a_{ij}b_{jk},$$ (1)

where $j$ is summed over for all possible values of $i$ and $k$. Therefore, in order for multiplication to be defined, the dimensions of the Matrices must satisfy

$$(n\times m)(m\times p) = (n\times p),$$ (2)

where $(a\times b)$ denotes a Matrix with $a$ rows and $b$ columns. Writing out the product explicitly,

$$\left[{\matrix{c_{11} & c_{12} & \cdots & c_{1p}\cr c_{21} &... ...ots & \vdots\cr b_{m1} & b_{m2} & \cdots & b_{mp}\cr}}\right],$$ (3)

where

$$\begin{eqnarray*} c_{11}&=&a_{11}b_{11}+a_{12}b_{21}+\ldots+a_{1m}b_{m1}\\ c_... ...{m2}\\ c_{np}&=&a_{n1}b_{1p}+a_{n2}b_{2p}+\ldots+a_{nm}b_{mp}. \end{eqnarray*}$$

Matrix multiplication is Associative, as can be seen by taking

$$[(ab)c]_{ij} = (ab)_{ik} c_{kj} = (a_{il}b_{lk})c_{kj}.$$ (4)

Now, since $a_{il}$, $b_{lk}$, and $c_{kj}$ are Scalars, use the Associativity of Scalar Multiplication to write

$$(a_{il}b_{lk})c_{kj} = a_{il}(b_{lk}c_{kj}) = a_{il}(bc)_{lj} = [a(bc)]_{ij}.$$ (5)

Since this is true for all $i$ and $j$, it must be true that

$$(ab)c = a(bc).$$ (6)

That is, matrix multiplication is Associative. However, matrix multiplication is not, in general, Commutative (although it is Commutative if ${\hbox{\sf A}}$ and ${\hbox{\sf B}}$ are Diagonal and of the same dimension).

The product of two Block Matrices is given by multiplying each block

$\left[{\matrix{o & o\cr o & o\cr & & o\cr & & & o & o & o\cr & & & o & o & o\cr... ...x\cr & & x\cr & & & x & x & x\cr & & & x & x & x\cr & & & x & x & x\cr}}\right]$
$=\left[{\matrix{\left[{\matrix{o & o\cr o & o\cr}}\right]\left[{\matrix{x & x\c... ...\right]\left[{\matrix{x & x & x\cr x & x & x\cr x & x & x\cr}}\right]}}\right].$
	(7)

See also Matrix, Matrix Addition, Matrix Inverse, Strassen Formulas

References

Arfken, G. Mathematical Methods for Physicists, 3rd ed. Orlando, FL: Academic Press, pp. 178-179, 1985.