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

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```Introduction to Matrix Algebra \u2013 Copyright \u2013 Autar K Kaw - 1-
Introduction
to
MATRIX

KAW
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-

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```