Autar Kaw   Introduction to Matrix Algebra (2008, www.autarkaw.com)
176 pág.

Autar Kaw Introduction to Matrix Algebra (2008, www.autarkaw.com)


DisciplinaTopicos de Algebra Elementar2 materiais48 seguidores
Pré-visualização24 páginas
Introduction to Matrix Algebra \u2013 Copyright \u2013 Autar K Kaw - 1- 
Introduction 
to 
MATRIX 
ALGEBRA© 
 
 
 
 
KAW 
© Copyrighted to Autar K. Kaw \u2013 2002 
Introduction to Matrix Algebra \u2013 Copyright \u2013 Autar K Kaw - 2- 
Introduction 
to 
MATRIX 
ALGEBRA 
 
 
 
 
 
Autar K. Kaw 
University of South Florida 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
Autar K. Kaw 
Professor & Jerome Krivanek Distinguished Teacher 
Mechanical Engineering Department 
University of South Florida, ENB 118 
4202 E. Fowler Avenue 
Tampa, FL 33620-5350. 
Office: (813) 974-5626 
Fax: (813) 974-3539 
E-mail: kaw@eng.usf.edu 
URL: http://www.eng.usf.edu/~kaw 
Introduction to Matrix Algebra \u2013 Copyright \u2013 Autar K Kaw - 3- 
Table of Contents 
 
Chapter 1: Introduction \u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026..\u2026 4 
 
Chapter 2: Vectors \u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026 16 
 
Chapter 3: Binary Matrix Operations \u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026 34 
 
Chapter 4: Unary Matrix Operations \u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026 47 
 
Chapter 5: System of Equations \u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026. 64 
 
Chapter 6: Gaussian Elimination \u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026 95 
 
Chapter 7: LU Decomposition \u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026.\u2026 116 
 
Chapter 8: Gauss- Siedal Method\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026 131 
 
Chapter 9: Adequacy of Solutions\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026 145 
 
Chapter 10: Eigenvalues and Eigenvectors \u2026\u2026\u2026\u2026\u2026\u2026\u2026\u2026.. 161 
Introduction to Matrix Algebra \u2013 Copyright \u2013 Autar K Kaw - 4- 
Chapter 1 
Introduction 
_________________________________ 
After reading this chapter, you should be able to 
§ Know what a matrix is 
§ Identify special types of matrices 
§ When two matrices are equal 
_________________________________ 
What is a matrix? 
Matrices are everywhere. If you have used a spreadsheet such as Excel or Lotus or 
written a table, you have used a matrix. Matrices make presentation of numbers clearer 
and make calculations easier to program. Look at the matrix below about the sale of tires 
in a Blowoutr\u2019us store \u2013 given by quarter and make of tires. 
 Quarter 1 Quarter 2 Quarter 3 Quarter 4 
Copper
Michigan
Tirestone
 
ê
ê
ê
ë
é
6
5
25
 
16
10
20
 
7
15
3
 
ú
ú
ú
û
ù
27
25
2
 
If one wants to know how many Copper tires were sold in Quarter 4, we go along the row 
\u2018Copper\u2019 and column \u2018Quarter 4\u2019 and find that it is 27. 
So what is a matrix? 
A matrix is a rectangular array of elements. The elements can be symbolic expressions or 
numbers. Matrix [A] is denoted by 
Introduction to Matrix Algebra \u2013 Copyright \u2013 Autar K Kaw - 5- 
[ ]
ú
ú
ú
ú
û
ù
ê
ê
ê
ê
ë
é
=
mnmm
n
n
aaa
aaa
aaa
A
.......
.......
.......
21
22221
11211
MM 
Row i of [A] has n elements and is [ ]inii aaa ....2 1 and 
Column j of [A] has m elements and is 
ú
ú
ú
ú
ú
û
ù
ê
ê
ê
ê
ê
ë
é
mj
j
j
a
a
a
M
2
1
 
Each matrix has rows and columns and this defines the size of the matrix. If a matrix [A] 
has m rows and n columns, the size of the matrix is denoted by m x n. The matrix [A] 
may also be denoted by [A]mxn to show that [ ]A is a matrix with m rows and n columns. 
Each entry in the matrix is called the entry or element of the matrix and is denoted by aij 
where i is the row number and j is the column number of the element. 
The matrix for the tire sales example could be denoted by the matrix [A] as 
[ ]
ú
ú
ú
û
ù
ê
ê
ê
ë
é
=
277166
2515105
232025
A 
There are 3 rows and 4 columns, so the size of the matrix is 3 x 4. In the above [ ]A 
matrix, 27a 34 = . 
What are the special types of matrices? 
Vector: A vector is a matrix that has only one row or one column. There are two types 
of vectors \u2013 row vectors and column vectors. 
Row vector: If a matrix has one row, it is called a row vector 
]bbb[]B[ m21 KK= 
and \u2018m\u2019 is the dimension of the row vector. 
_________________________________ 
Introduction to Matrix Algebra \u2013 Copyright \u2013 Autar K Kaw - 6- 
Example 
Give an example of a row vector. 
Solution 
[B] = [25 20 3 2 0] is an example of a row vector of dimension 5. 
_________________________________ 
Column vector: If a matrix has one column, it is called a column vector 
[ ]
ú
ú
ú
ú
û
ù
ê
ê
ê
ê
ë
é
=
n
1
c
c
C M
M
 
and n is the dimension of the vector. 
_________________________________ 
Example 
Give an example of a column vector. 
Solution 
 [ ]
ú
ú
ú
û
ù
ê
ê
ê
ë
é
=
6
5
25
C is an example of a column vector 
of dimension 3. 
_________________________________ 
Submatrix: If some row(s) or/and column(s) of a matrix [A] are deleted, the remaining 
matrix is called a submatrix of [A]. 
Example 
Find some of the submatrices of the matrix 
Introduction to Matrix Algebra \u2013 Copyright \u2013 Autar K Kaw - 7- 
[ ] ú
û
ù
ê
ë
é
-
=
213
264
A 
Solution 
[ ] [ ] ú
û
ù
ê
ë
é
ú
û
ù
ê
ë
é
-úû
ù
ê
ë
é
- 2
2
,4,264,
13
64
,
213
264
are all submatrices of [A]. Can you find other 
submatrices of [A]? 
_________________________________ 
Square matrix: If the number of rows (m) of a matrix is equal to the number of columns 
(n) of the matrix, (m = n), it is called a square matrix. The entries a11, a22, . . . ann are 
called the diagonal elements of a square matrix. Sometimes the diagonal of the matrix is 
also called the principal or main of the matrix. 
_________________________________ 
Example 
Give an example of a square matrix. 
Solution 
[ ]
ú
ú
ú
û
ù
ê
ê
ê
ë
é
=
7156
15105
32025
A 
is a square matrix as it has same number of rows and columns, that is, three. 
The diagonal elements of [A] are a11 = 25, a22 = 10, a33 = 7. 
_________________________________ 
Upper triangular matrix: A mxn matrix for which aij = 0, i>j is called an upper 
triangular matrix. That is, all the elements below the diagonal entries are zero. 
_________________________________ 
Introduction to Matrix Algebra \u2013 Copyright \u2013 Autar K Kaw - 8- 
Example 
Give an example of an upper triangular matrix. 
Solution 
[ ]
ú
ú
ú
û
ù
ê
ê
ê
ë
é
-
-
=
1500500
6001.00
0710
A 
is an upper triangular matrix. 
_________________________________ 
Lower triangular matrix: A mxn matrix for which aij = 0, j > i is called a lower 
triangular matrix. That is, all the elements above the diagonal entries are zero. 
_________________________________ 
Example 
Give an example of a lower triangular matrix. 
Solution 
[ ]
ú
ú
ú
û
ù
ê
ê
ê
ë
é
=
15.26.0
013.0
001
A 
is a lower triangular matrix. 
_________________________________ 
Diagonal matrix: A square matrix with all non-diagonal elements equal to zero is called 
a diagonal matrix, that is, only the diagonal entries of the square matrix can be non-zero, 
(aij = 0, i ¹ j). 
_________________________________ 
Example 
Introduction to Matrix Algebra \u2013 Copyright \u2013 Autar K Kaw - 9- 
Give examples of a diagonal matrix. 
Solution 
[ ] =A
ú
ú
ú
û
ù
ê
ê
ê
ë
é
500
01.20
003
 
is a diagonal matrix. 
Any or all the diagonal entries of a diagonal matrix can be zero. 
For example 
[ ]
ú
ú
ú
û
ù
ê
ê
ê
ë
é
=
000
01.20
003
A 
is also a diagonal matrix. 
_________________________________ 
Identity matrix: A diagonal matrix with all diagonal elements equal to one is called an 
identity matrix, (aij = 0, i ¹ j; and aii = 1 for all i). 
. 
_________________________________ 
Example 
Give an example of an identity matrix. 
Solution