The Mathematics of Financial Modelingand Investment Management

5-Matrix Algebra Page 161 Wednesday, February 4, 2004 12:49 PM

Matrix Algebra 161

a 11 , – λ · a 1 , j · a 1 , n x 1

· · · · · ·

(A – λI)x = ai, 1 · aii, – λ · ain, xi = 0

· · · · · ·

an, 1 · anj, · ann, – λ xn

This system of equations has nontrivial solutions only if the matrix A –

λI is singular. To determine the eigenvectors and the eigenvalues of the

matrix A we must therefore solve the equation

a 11 , – λ · a 1 , j · a 1 , n

· · · · ·

A – λI = ai, 1 · aii, – λ · ain, = 0

· · · · ·

an, 1 · anj, · ann, – λ

The expansion of this determinant yields a polynomial φ(λ) of

degree n known as the characteristic polynomial of the matrix A. The

equation φ(λ) = 0 is known as the characteristic equation of the matrix

A. In general, this equation will have n roots λs which are the eigenval-

ues of the matrix A. To each of these eigenvalues corresponds a solution

of the system of linear equations as illustrated below:

a 11 , – λs· a 1 , j · a 1 , n x (^1) s
· · · · · ·
ai, 1 · aii, – λs · ain, xis = 0
· · · · · ·
an, 1 · anj, · ann, – λs xn
s
Each solution represents the eigenvector xs corresponding to the eigen-
vector λs. As we will see in Chapter 12, the determination of eigenvalues
and eigenvectors is the basis for principal component analysis.
DIAGONALIZATION AND SIMILARITY
Diagonal matrices are much easier to handle than fully populated matri -
ces. It is therefore important to create diagonal matrices equivalent (in a
sense to be precisely defined) to a given matrix. Consider two square