Page 124 - Numerical Analysis Using MATLAB and Excel

Chapter 4 Matrices and Determinants

In an upper triangular matrix, not all elements above the diagonal need to be non−zero. For
applications, refer to Chapter 14.

† A square matrix is said to be lower triangular, when all the elements above the diagonal are
zero. The matrix below is a lower triangular matrix. For applications, refer to Chapter 14.
B

a 11 0 0 … 0
a 21 a 22 0 … 0
B = ……… 0 0 (4.5)
……… … 0
a m1 a m2 a m3 … a mn

In a lower triangular matrix, not all elements below the diagonal need to be non−zero.

† A square matrix is said to be diagonal, if all elements are zero, except those in the diagonal. The
matrix below is a diagonal matrix.
C

a 11 0 0 … 0
0a 22 0 … 0
C = 0 0 … 0 0 (4.6)
0 0 0 … 0
0 0 0 … a mn

k
† A diagonal matrix is called a scalar matrix, if a 11 = a 22 = a 33 = … = a nn = k where is a sca-
lar. The matrix below is a scalar matrix with k = . 4
D

4 000
D = 04 0 0 (4.7)
00 4 0
00 0 4

I
A scalar matrix with k = 1 , is called an identity matrix . Shown below are 2 × 2 , 3 × 3 , and
4 × 4 identity matrices.

1 0 0 0
10 0
10 01 0 0 (4.8)
01 01 0 00 1 0
00 1
00 0 1

The MATLAB eye(n) function displays an n × n identity matrix. For example,

4−6 Numerical Analysis Using MATLAB® and Excel®, Third Edition

Copyright © Orchard Publications