Page 13 - Vector Analysis

§1.4 Matrices 9

where aij P F is called the (i, j)-th entry of A, and is denoted by [A]ij. We write A =
[aij]1ďiďm;1ďjďn or simply A = [aij]mˆn to denote that A is an m ˆ n matrix whose (i, j)-th
entry is aij. A is called a square matrix if m = n. The 1 ˆ m matrix

[]
ai˚ = ai1 ai2 ¨ ¨ ¨ ain

is called the i-th row of A, and the m ˆ 1 matrix


a1j
a˚j =  a2j 
...

amj

is called the j-th column of A.

Definition 1.29 (Matrix addition). Let A = [aij]mˆn and B = [bij]mˆn be two m ˆ n

matrices over a scalar field F. The sum of A and B, denoted by A + B, is another m ˆ n

matrix defined by A + B = [aij + bij]mˆn or more precisely,


a11 + b11 a12 + b12 ¨ ¨ ¨ a1n + b1n
A + B =   .
a21 + b21 a22 + b22 ¨¨¨ a2n + b2n
... ... ... ...

am1 + bm1 am2 + bm2 ¨ ¨ ¨ amn + bmn

Definition 1.30 (Scalar multiplication). Let A = [aij]mˆn be an m ˆ n matrix over a scalar
field F, and α P F. The scalar multiplication of α and A, denoted by αA, is an m ˆ n matrix

defined by αA = [αaij]mˆn or more precisely,


αa11 αa12 ¨ ¨ ¨ αa1n
αA =   .
αa21 αa22 ¨¨¨ αa2n
... ... ... ...

αam1 αam2 ¨ ¨ ¨ αamn

Proposition 1.31. The space M(m, n; F) is a vector space over F under the matrix addition
and scalar multiplication defined in previous two definitions.

Definition 1.32 (Matrix product). Let A P M(m, n; F) and B P M(n, ℓ; F) be two matrices

over a scalar field F. The matrix product of A and B, denoted by AB, is an m ˆ ℓ matrix
n

given by AB = [cij]mˆn with cij = ř aikbkj. In other words, the (i, j)-th entry of the

k=1

product AB is the inner product of the i-th row of A and the j-th column of B.