The Mathematics of Financial Modelingand Investment Management

5-Matrix Algebra Page 151 Wednesday, February 4, 2004 12:49 PM

Matrix Algebra 151

LINEAR INDEPENDENCE AND RANK

Consider an n×m matrix A. A set of p columns extracted from the

matrix A

·a 1 , i 1 ·a 1 , ip ·

· · · · ·

· · · · ·

· · · · ·

·ani, 1 ·ani, p ·

are said to be linearly independent if it is not possible to find p constants

βs, s = 1,...,p such that the following n equations are simultaneously sat-

isfied:

β 1 a 1 , i +

1

+ ...βpa 1 , i

p

= 0

........................

β 1 ani, 1 + ...β+ pani, p = 0

Analogously, a set of q rows extracted from the matrix A are said to

be linearly independent if it is not possible to find q constants λs, s =

1,...,q, such that the following m equations are simultaneously satisfied:

λ 1 ai 1 , 1 + ...λ+ qaiq, 1 = 0

........................

λ 1 ai 1 , m + ...λ+ qaiq, m = 0

It can be demonstrated that in any matrix the number p of linearly

independent columns is the same as the number q of linearly indepen-

dent rows. This number is equal, in turn, to the rank r of the matrix.

Recall that an n×m matrix A is said to be of rank r if at least one of its

(square) r-minors is different from zero while all (r+1)-minors, if any,

are zero. The constant, p, is the same for rows and for columns. We can

now give an alternative definition of the rank of a matrix:

Given an n×m matrix A, its rank, denoted rank(A), is the number r of

linearly independent rows or columns. This definition is meaningful

because the row rank is always equal to the column rank.