The Mathematics of Financial Modelingand Investment Management

5-Matrix Algebra Page 148 Wednesday, February 4, 2004 12:49 PM

148 The Mathematics of Financial Modeling and Investment Management

Upper and Lower Triangular Matrix

A matrix A is called upper triangular if aij = 0, i > j. In other words, an

upper triangular matrix is a matrix whose elements in the triangle below

the diagonal are all zero as is illustrated below:

a 11 , ·a 1 , i· a 1 , n

· ··· ·

A = 0· aii, · ain, [upper triangular]

· ··· ·

0·0· ann,

A matrix A is called lower triangular if aij = 0, i < j. In other words,

a lower triangular matrix is a matrix whose elements in the triangle

above the diagonal are zero as is illustrated below:

a 11 , ·0· 0

·····

A = ·· aii, ·0 [lower triangular]

· ··· ·

an, 1 ·ani, · ann,

DETERMINANTS

Consider a square, n×n, matrix A. The determinant of A, denoted A , is

defined as follows:

n

= ∑(– 1 )

tj( 1 , ..., jn)

A ∏ aij

i = 1

where the sum is extended over all permutations (j 1 ,...,jn) of the set (1,

2,...,n) and t(j 1 ,...,jn) is the number of transpositions (or inversions of

positions) required to go from (1,2,...,n) to (j 1 ,...,jn).

Otherwise stated, a determinant is the sum of all different products

formed taking exactly one element from each row with each product

multiplied by

(– 1 )

tj( 1 , ..., jn )