Introduction to Matrix Algebra

176 Pages·2008·0.46 MB·English

by  Autar Kaw

Introduction
to
MATRIX
ALGEBRA
©

KAW
© Copyrighted to Autar K. Kaw – 2002
Introduction to Matrix Algebra – Copyright – Autar K Kaw - 1-
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: [phone]
Fax: [phone]
E-mail: [email protected]
URL: http://www.eng.usf.edu/~kaw
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Table of Contents

Chapter 1: Introduction …………………………………………..… 4

Chapter 2: Vectors ………………………………………………… 16

Chapter 3: Binary Matrix Operations …………………………… 34

Chapter 4: Unary Matrix Operations …………………………… 47

Chapter 5: System of Equations …………………………………. 64

Chapter 6: Gaussian Elimination ………………………………… 95

Chapter 7: LU Decomposition ………………………………….… 116

Chapter 8: Gauss- Siedal Method………………………………… 131

Chapter 9: Adequacy of Solutions………………………………… 145

Chapter 10: Eigenvalues and Eigenvectors …………………….. 161
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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
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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’us store – given by quarter and make of tires.
Quarter 1 Quarter 2 Quarter 3 Quarter 4
Tirestone Ø 25 20 3 2 ø
Œ œ
Michigan 5 10 15 25
Œ œ
Copper μ 6 16 7 27ϧ
If one wants to know how many Copper tires were sold in Quarter 4, we go along the row
‘Copper’ and column ‘Quarter 4’ 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
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Ø a a ....... a ø
Œ 11 12 1n œ
[ ] Œ a a ....... a œ
A = 21 22 2n
Œ œ
M M
Œ œ
º a a ....... a ß
m1 m2 mn
[ ]
Row i of [A] has n elements and is a a ....a and
i1 i2 in
Ø a ø
Œ 1jœ
a
Œ œ
Column j of [A] has m elements and is 2j
Œ œ
M
Œ œ
μ a ϧ
mj
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] to show that A is a matrix with m rows and n columns.
mxn
Each entry in the matrix is called the entry or element of the matrix and is denoted by a
ij
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
Ø 25 20 3 2 ø
[ ] Œ œ
A = 5 10 15 25
Œ œ
Œ œ
º 6 16 7 27ß
[ ]
There are 3 rows and 4 columns, so the size of the matrix is 3 x 4. In the above A
matrix, a = 27.
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 – row vectors and column vectors.
Row vector: If a matrix has one row, it is called a row vector
[B]=[b b b ]
1 2KK m
and ‘m’ is the dimension of the row vector.
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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.
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Column vector: If a matrix has one column, it is called a column vector
Ø c ø
Œ 1œ
[C]= Œ M œ
Œ œ
M
Œ œ
º c ß
n
and n is the dimension of the vector.
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Example
Give an example of a column vector.
Solution
Ø 25ø
[ ] Œ œ
C = 5 is an example of a column vector
Œ œ
Œ œ
º 6 ß
of dimension 3.
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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
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[ ] Ø 4 6 2ø
A = Œ œ
º 3 - 1 2ß
Solution
Ø 4 6 2ø Ø 4 6 ø [ ][ ] Ø 2ø
Œ œ ,Œ œ , 4 6 2, 4,Œ œ are all submatrices of [A]. Can you find other
º 3 - 1 2ß º 3 - 1ß º 2ß
submatrices of [A]?
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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 a , a , . . . a are
11 22 nn
called the diagonal elements of a square matrix. Sometimes the diagonal of the matrix is
also called the principal or main of the matrix.
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Example
Give an example of a square matrix.
Solution
Ø 25 20 3ø
[ ] Œ œ
A = 5 10 15
Œ œ
Œ œ
º 6 15 7ß
is a square matrix as it has same number of rows and columns, that is, three.
The diagonal elements of [A] are a = 25, a = 10, a = 7.
11 22 33
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Upper triangular matrix: A mxn matrix for which a = 0, i>j is called an upper
ij
triangular matrix. That is, all the elements below the diagonal entries are zero.
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Example
Give an example of an upper triangular matrix.
Solution
Ø 10 - 7 0 ø
[ ] Œ œ
A = 0 - 0.001 6
Œ œ
Œ œ
º 0 0 15005ß
is an upper triangular matrix.
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Lower triangular matrix: A mxn matrix for which a = 0, j > i is called a lower
ij
triangular matrix. That is, all the elements above the diagonal entries are zero.
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Example
Give an example of a lower triangular matrix.
Solution
Ø 1 0 0ø
[ ] Œ œ
A = 0.3 1 0
Œ œ
Œ œ
º 0.6 2.5 1ß
is a lower triangular matrix.
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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,
(a = 0, i „ j).
ij
_________________________________
Example
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Give examples of a diagonal matrix.
Solution
Ø 3 0 0ø
[ ] Œ œ
A = 0 2.1 0
Œ œ
Œ œ
º 0 0 5ß
is a diagonal matrix.
Any or all the diagonal entries of a diagonal matrix can be zero.
For example
Ø 3 0 0ø
[ ] Œ œ
A = 0 2.1 0
Œ œ
Œ œ
º 0 0 0ß
is also a diagonal matrix.
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Identity matrix: A diagonal matrix with all diagonal elements equal to one is called an
identity matrix, (a = 0, i „ j; and a = 1 for all i).
ij ii
.
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Example
Give an example of an identity matrix.
Solution
Ø 1 0 0 0ø
Œ œ
0 1 0 0
Œ œ
[A] =
Œ œ
0 0 1 0
Œ œ
º 0 0 0 1ß
is an identity matrix.
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_________________________________
Zero matrix: A matrix whose all entries are zero is called a zero matrix, (a = 0 for all i
ij
and j).
_______________________________
Example
Give examples of a zero matrix.
Solution
Ø 0 0 0ø
[ ] Œ œ
A = 0 0 0
Œ œ
Œ œ
º 0 0 0ß
[ ] Ø 0 0 0 ø
B = Œ œ
º 0 0 0 ß
Ø 0 0 0 0ø
[ ] Œ œ
C = 0 0 0 0
Œ œ
Œ œ
º 0 0 0 0ß
[ ] [ ]
D = 0 0 0
are all examples of a zero matrix.
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Tridiagonal matrices: A tridiagonal matrix is a square matrix in which all elements not
on the major diagonal, the diagonal above the major diagonal and the diagonal below the
major diagonal are zero.
_________________________________
Example
Give an example of a tridiagonal matrix.
Solution
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