The Mathematics of Financial Modelingand Investment Management

5-Matrix Algebra Page 145 Wednesday, February 4, 2004 12:49 PM

Matrix Algebra 145

where the subscripts nm are the dimensions of the matrix.

The elements of a matrix can be either real numbers or complex

numbers. In the following, we will assume that elements are real num-

bers unless explicitly stated otherwise. If the matrix entries are real

numbers, the matrix is called a real matrix; if the aij are complex num-

bers, the matrix is called a complex matrix.

Two matrices are said to be equal if they are of the same dimensions

and have the same elements. Consider two matrices A = {aij}nm and B =

{bij}nm of the same order n×m:

A = B means {}aijnm = {}bij nm

Vectors are matrices with only one column or only one row. An n-

dimensional row vector is an n×1 matrix, an n-dimensional column vec-

tor is a 1×n matrix. A matrix can be thought of as an array of vectors.

Denote by aj the column vector formed by the j-th column of the matrix

A. The matrix A can then be written as A = []aj. This notation can be

generalized. Suppose that the two matrices B, C have the same number

n of rows and mB, mC columns respectively. The matrix A = [B C] is the

matrix whose first mB columns are formed by the matrix B and the fol-

lowing mC columns are formed by the matrix C.

SQUARE MATRICES

There are several types of matrices. First there is a broad classification

of square and rectangular matrices. A rectangular matrix can have dif-

ferent numbers of rows and columns; a square matrix is a rectangular

matrix with the same number n of rows as of columns.

Diagonals and Antidiagonals

An important concept for a square matrix is the diagonal. The diagonal

includes the elements that run from the first row, first column to the last

row, last column. For example, consider the following square matrix:

a 11 , ·a 1 ,j· a 1 ,n

· ··· ·

A = ai, 1 ·aij, · ain,

· ··· ·

an, 1 ·anj, · ann,

The diagonal terms are the aj,j terms.