The Mathematics of Financial Modelingand Investment Management

5-Matrix Algebra Page 146 Wednesday, February 4, 2004 12:49 PM

146 The Mathematics of Financial Modeling and Investment Management

The antidiagonals of a square matrix are the other diagonals that do

not run from the first row, first column to the last row, last column. For

example, consider the following 4×4 square matrix:

5 9 14 8

2 6 12 11

17 21 42 2

19 73 7 8

The diagonal terms include 5, 6, 42, 8. One antidiagonal is 2, 9. Another

antidiagonal is 17, 6, 14. Note that there are antidiagonal terms in rect-

angular matrices.

Identity Matrix

The n×n identity matrix, indicated as the matrix In, is a square matrix

whose diagonal elements (i.e., the entries with the same row and column

suffix) are equal to one while all other entries are zero:

1 0···0

0 1···0

I = ··· ·

n ·· · ·

·· ··

0 0···1

A matrix whose entries are all zero is called a zero matrix.

Diagonal Matrix

A diagonal matrix is a square matrix whose elements are all zero except

the ones on the diagonal:

a 11 0 ··· 0

0 a 22 ··· 0

A = · · · ·

· · · ·

· · · ·

0 0 ··· ann

Given a square n×n matrix A, the matrix dg A is the diagonal matrix

extracted from A. The diagonal matrix dg A is a matrix whose elements