Chapter 3 Solution of Linear Systems - Math/CS

48 CHAPTER 3. SOLUTION OF LINEAR SYSTEMS
(a)
(b)
(c)
(−1)x1 + 0x2 + 0x3 = −1
0x1 + 3x2 + 0x3 = 6
0x1 + 0x2 + (−5)x3 = −15
(−1)x1 + 0x2 + 0x3 = −1
2x1 + 3x2 + 0x3 = 8
(−1)x1 + 4x2 + (−5)x3 = −8
(−1)x1 + 2x2 + (−1)x3 = 0
0x1 + 3x2 + 6x3 = 24
0x1 + 0x2 + (−5)x3 = −15
⇔
⇔
⇔
⎡
⎣
⎡
⎣
⎡
⎣
−1 0 0
0 3 0
0 0 −5
−1 0 0
2 3 0
−1 4 −5
−1 2 −1
0 3 6
0 0 −5
Figure 3.1: Simply Solved Linear Systems
⎤ ⎡
⎦ ⎣
⎤ ⎡
⎦ ⎣
⎤ ⎡
⎦ ⎣
(c) the strictly upper triangular entries, i.e., the entries ai,j for which i < j.
x1
x2
x3
x1
x2
x3
x1
x2
x3
⎤
⎡
⎦ = ⎣
⎤
⎡
⎦ = ⎣
⎤
⎡
⎦ = ⎣
The locations of the strictly lower triangular, diagonal, and strictly upper triangular entries of A
are illustrated in Fig. 3.2. The lower triangular entries are composed of the strictly lower triangular
and diagonal entries, as illustrated in Fig. 3.3. Similarly, the upper triangular entries are composed
of the strictly upper triangular and diagonal entries.
a 11
a 21
a n1
a 12
(strictly lower triangular part)
(strictly upper triangular part)
(diagonal part) a nn
a n,n−1
a 1n
a n−1,n
Figure 3.2: Illustration of strict triangular and diagonal matrix entries.
Problem 3.1.7. Let A be a matrix of order n. Show that A has a total of n2 entries. Show that
n2 − n = n(n − 1) entries lie off the diagonal, and show that each strictly triangular portion of A
n(n − 1)
has entries.
2
Diagonal Linear Systems
A matrix A of order n is diagonal if all its nonzero entries are on its diagonal. (Note that this
description of a diagonal matrix does not state that the entries on the diagonal are nonzero.) A
diagonal linear system of equations of order n is one whose coefficient matrix is diagonal.
−1
6
−15
−1
8
−8
0
24
−15
⎤
⎦
⎤
⎦
⎤
⎦