Matrix Algebra for Engineers

2018·5.2 MB·english

by  Jeffrey Chasnov

JEFFREY CHASNOV
MATRIX ALGEBRA
FOR ENGINEERS
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Matrix Algebra for Engineers
1st edition
© 2018 Jeffrey Chasnov & bookboon.com
ISBN 978-87-403-2383-2
Peer review by Prof Shingyu Leung
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MATRIX ALGEBRA FOR ENGINEERS Contents
CONTENTS
Preface 7
Week I: Matrices 8
1 Definition of a matrix 10
2 A ddition and multiplication of matrices 12
3 Special matrices 14
4 Transpose matrix 16
5 Inner and outer products 18
6 Inverse matrix 19
7 Orthogonal matrices 21
8 O rthogonal matrices example 22
9 Permutation matrices 24
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MATRIX ALGEBRA FOR ENGINEERS Contents
Week II: Systems of linear equations 25
10 Gaussian elimination 27
11 Reduced row echelon form 30
12 Computing inverses 32
13 Elementary matrices 34
14 LU decomposition 36
15 Solving (LU)x = b 38
Week III: Vector spaces 40
16 Vector spaces 42
17 Linear independence 44
18 Span, basis and dimension 46
19 Gram-Schmidt process 47
20 G ram-Schmidt process example 48
21 Null space 50
22 A pplication of the null space 52
23 Column space 54
24 R ow space, left null space and rank 56
25 Orthogonal projections 58
26 The least-squares problem 60
27 S olution of the least-squares problem 62
Week IV: Eigenvalues and eigenvectors 64
28 T wo-by-two and three-by-three determinants 66
29 Laplace expansion 68
30 Leibniz formula 70
31 Properties of a determinant 72
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MATRIX ALGEBRA FOR ENGINEERS Contents
32 The eigenvalue problem 74
33 F inding eigenvalues and eigenvectors (1) 76
34 F inding eigenvalues and eigenvectors (2) 78
35 Matrix diagonalization 79
36 M atrix diagonalization example 80
37 Powers of a matrix 82
38 Powers of a matrix example 83
Appendix A Problem solutions 84
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MATRIX ALGEBRA FOR ENGINEERS PrefaCe
PREFACE
These are my lecture notes for my online Coursera course, Matrix Algebra for Engineers. I
have divided these notes into chapters called Lectures, with each Lecture corresponding to
a video on Coursera. I have also uploaded all my Coursera videos to YouTube, and links
are placed at the top of each Lecture.
There are problems at the end of each lecture chapter and I have tried to choose problems
that exemplify the main idea of the lecture. Students taking a formal university course in
matrix or linear algebra will usually be assigned many more additional problems, but here I
follow the philosophy that less is more. I give enough problems for students to solidify their
understanding of the material, but not too many problems that students feel overwhelmed
and drop out. I do encourage students to attempt the given problems, but if they get stuck,
full solutions can be found in the Appendix.
The mathematics in this matrix algebra course is at the level of an advanced high school
student, but typically students would take this course after completing a university-level
single variable calculus course. There are no derivatives and integrals in this course, but
student’s are expected to have a certain level of mathematical maturity. Nevertheless, anyone
who wants to learn the basics of matrix algebra is welcome to join.
Jeffrey R. Chasnov
Hong Kong
July 2018
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WEEK I: MATRICES
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In this week’s lectures, we learn about matrices. Matrices are rectangular arrays
of numbers or other mathematical objects and are fundamental to engineering
mathematics. We will define matrices and how to add and multiply them,
discuss some special matrices such as the identity and zero matrix, learn about
transposes and inverses, and define orthogonal and permutation matrices.
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MATRIX ALGEBRA FOR ENGINEERS Definition of a matrix
1 DEFINITION OF A MATRIX
View this lecture on YouTube
An m-by-n matrix is a rectangular array of numbers (or other mathematical objects) with
m rows and n columns. For example, a two-by-two matrix A, with two rows and two
columns, looks like
a b
A= .
(cid:31)c d(cid:30)
The first row has elements a and b, the second row has elements c and d. The first column
has elements a and c; the second column has elements b and d. As further examples, two-
by-three and three-by-two matrices look like
a d
a b c
B= , C= ⎛b e⎞.
�d e f�
c f
⎜ ⎟
⎝ ⎠
Of special importance are column matrices and row matrices. These matrices are also called
vectors. The column vector is in general n-by-one and the row vector is one-by-n. For
example, when n = 3, we would write a column vector as
a
x= ⎛b⎞,
c
⎜ ⎟
⎝ ⎠
and a row vector as
y= a b c .
(cid:31) (cid:30)
A useful notation for writing a general m-by-n matrix A is
a a a
11 12 ··· 1n
a a a
A= ⎛⎜ 2...1 2...2 ·.·..· 2...n⎞⎟.
⎜ ⎟
⎜a a a ⎟
⎜ m1 m2 ··· mn⎟
⎝ ⎠
Here, the matrix element of A in the ith row and the jth column is denoted as a .
ij
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