The Mathematics of Financial Modelingand Investment Management

5-Matrix Algebra Page 150 Wednesday, February 4, 2004 12:49 PM

150 The Mathematics of Financial Modeling and Investment Management

formed with the coefficients of the variables is called the coefficient

matrix. The terms bi are called the constant terms. The augmented

matrix [A b]—formed by adding to the coefficient matrix a column

formed with the constant term—is represented below:

a 11 , · a 1 , j · a 1 , mb 1

· ··· ·

[Ab] = ai, 1 · aij, · aim, bi

· ··· ·

an, 1 ·anj, · anm, bn

If the constant terms on the right side of the equations are all zero, the

system is called homogeneous. If at least one of the constant terms is dif-

ferent from zero, the system is called nonhomogeneous. A system is called

consistent if it admits a solution, i.e., if there is a set of values of the vari-

ables that simultaneously satisfy all the equations. A system is called

inconsistent if there is no set of numbers that satisfy the system equations.

Let’s first consider the case of nonhomogeneous linear systems. The

fundamental theorems of linear systems state that:

■ Theorem 1. A system of n linear equations in m unknowns is consistent

(i.e., it admits a solution) if and only if the coefficient matrix and the

augmented matrix have the same rank.

■ Theorem 2. If a consistent system of n equations in m variables is of

rank r < m, it is possible to choose n–r unknowns so that the coefficient

matrix of the remaining r unknowns is of rank r. When these m–r vari-

ables are assigned any arbitrary value, the value of the remaining vari-

ables is uniquely determined.

An immediate consequence of the fundamental theorems is that (1) a

system of n equations in n unknown variables admits a solution and (2) the

solution is unique if and only if both the coefficient matrix and the aug-

mented matrix are of rank n.

Let’s now examine homogeneous systems. The coefficient matrix and

the augmented matrix of a homogeneous system always have the same

rank and thus a homogeneous system is always consistent. In fact, the

trivial solution x 1 = ... = xm = 0 always satisfies a homogeneous system.

Consider now a homogeneous system of n equations in n unknowns.

If the rank of the coefficient matrix is n, the system has only the trivial

solution. If the rank of the coefficient matrix is r < n, then Theorem 2

ensures that the system has a solution other than the trivial solution.