Download Matrix product. Let A be an m × n matrix. If x ∈ IR is a

Matrix product. Let A be an m × n matrix. If x ∈ IRn is a (column)
vector, then
n
X
aik xk = bi , 1 ≤ i ≤ m
Ax = b :
k=1
defines the matrix product function f (x) = Ax from IRn to IRm . Similarly,
if B is an n × p matrix and y ∈ IRp is a (column) vector, then
p
X
By = x :
bkj yj = xk , 1 ≤ k ≤ n
j=1
defines the matrix product function g(y) = By from IRp to IRn . Then the
composite function f ◦ g from IRp to IRm is given by
p
p X
n
n
X
X
X
aik bkj )yj .
bkj yj =
(
aik
f (g(y)) = A(By) =
k=1
j=1
j=1 k=1
If we define the matrix product AB by
n
X
(AB)ij =
aik bkj ,
k=1
then we have
(1)
f (g(y)) = (AB)y,
and so composition of matrix product functions is consistent with matrix
multiplication.
 
~a1
 ~a2 

Write the matrix A as a column of rows A = 
. . . and the matrix B
~am
as a row of columns B = b1 b2 . . . bp . The ij term in the matrix
product is given by the row-column product, (AB)ij = ~ai bj , 1 ≤ i ≤
m, 1 ≤ j ≤ p.
1