The Mathematics of Financial Modelingand Investment Management

5-Matrix Algebra Page 160 Wednesday, February 4, 2004 12:49 PM

160 The Mathematics of Financial Modeling and Investment Management

The adjoint of a matrix A is therefore the transpose of the matrix

obtained by replacing the elements of A with their cofactors.

If the matrix A is nonsingular, and therefore admits an inverse, it

can be demonstrated that

A

• 1 Adj () A
  = ------------------
  A

A square matrix A of order n is said to be orthogonal if the follow-

ing property holds:

AA′ = A′A = In

Because in this case A must be of full rank, the transpose of an orthogo-

nal matrix coincides with its inverse: A–1 = A′.

EIGENVALUES AND EIGENVECTORS

Consider a square matrix A of order n and the set of all n-dimensional

vectors. The matrix A is a linear operator on the space of vectors. This

means that A operates on each vector producing another vector and that

the following property holds:

A(ax + by) = aAx + bAy

Consider now the set of vectors x such that the following property

holds:

Ax = λx

Any vector such that the above property holds is called an eigenvector

of the matrix A and the corresponding value of λis called an eigenvalue.

To determine the eigenvectors of a matrix and the relative eigenval-

ues, consider that the equation Ax = λx can be written as follows:

(A – λI)x = 0

which can, in turn, be written as a system of linear equations: