The Mathematics of Financial Modelingand Investment Management

5-Matrix Algebra Page 162 Wednesday, February 4, 2004 12:49 PM

162 The Mathematics of Financial Modeling and Investment Management

matrices A and B. The matrices A and B are called similar if there exists

a nonsingular matrix R such that

B = R–1AR

The following two theorems can be demonstrated:

■ Theorem 1. Two similar matrices have the same eigenvalues.

■ Theorem 2. If yi is an eigenvector of the matrix B = R–1AR corre-

sponding to the eigenvalue λ i, then the vector xi = Ryi is an eigenvector

of the matrix A corresponding to the same eigenvalue λ i.

A diagonal matrix of order n always has n linearly independent eigen-

vectors. Consequently, a square matrix of order n has n linearly inde-

pendent eigenvectors if and only if it is similar to a diagonal matrix.

Suppose the square matrix of order n has n linearly independent

eigenvectors xi and n distinct eigenvalues λ i. This is true, for instance, if

A is a real, symmetric matrix of order n. Arrange the eigenvectors,

which are column vectors, in a square matrix: P = {xi}. It can be demon-

strated that P–1AP is a diagonal matrix where the diagonal is made up of

the eigenvalues:

λ 1 0000

0 · 000

P

• 1
  AP = 00 λ i 00
  000·0
  0000 λ n

SINGULAR VALUE DECOMPOSITION

Suppose that the n× m matrix A with m ≥ n has rank(A) = r > 0. It can be

demonstrated that there exists three matrices U, W, V such that the fol-

lowing decomposition, called singular value decomposition, holds:

A = UWV′′′′

and such that U is n× r with U′′′′U = Ir; W is diagonal, with non-negative

diagonal elements; and V is m× r with V′′′′V = Ir.