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TABLE OF CONTENTS
1. Linear Equations ................1
1.1 Introduction .........................1
1.2 Gaussian Elimination and Matrices .............3
1.3 Gauss–Jordan Method .................. 15
1.4 Two-Point Boundary Value Problems ........... 18
1.5 Making Gaussian Elimination Work ............ 21
1.6 Ill-Conditioned Systems .................. 33
2. Rectangular Systems and Echelon Forms ..... 41
2.1 Row Echelon Form and Rank ............... 41
2.2 Reduced Row Echelon Form ................ 47
2.3 Consistency of Linear Systems ............... 53
2.4 Homogeneous Systems .................. 57
2.5 Nonhomogeneous Systems ................. 64
2.6 Electrical Circuits ..................... 73
3. Matrix Algebra ................ 79
3.1 From Ancient China To Arthur Cayley .......... 79
3.2 Addition and Transposition ................ 81
3.3 Linearity ......................... 89
3.4 Why Do It This Way ................... 93
3.5 Matrix Multiplication ................... 95
3.6 Properties of Matrix Multiplication ............105
3.7 Matrix Inversion .....................115
3.8 Inverses of Sums and Sensitivity ..............124
3.9 Elementary Matrices and Equivalence ...........131
3.10 The LU Factorization ..................141
4. Vector Spaces .................159
4.1 Spaces and Subspaces ...................159
4.2 Four Fundamental Subspaces ...............169
4.3 Linear Independence ...................181
4.4 Basis and Dimension ...................194
4.5 More about Rank .....................210
4.6 Classical Least Squares ..................221
4.7 Linear Transformations ..................236
4.8 Change of Basis and Similarity ..............249
4.9 Invariant Subspaces ....................257
5. Norms, Inner Products, and Orthogonality ....267
5.1 Vector Norms .......................267
5.2 Matrix Norms .......................277
5.3 Inner-Product Spaces ...................284
5.4 Orthogonal Vectors ....................292
5.5 Gram–Schmidt Procedure .................306
5.6 Unitary and Orthogonal Matrices .............319
5.7 Orthogonal Reduction ...................340
5.8 Discrete Fourier Transform ................355
5.9 Complementary Subspaces ................382
5.10 Range-Nullspace Decomposition .............393
5.11 Orthogonal Decomposition ................402
5.12 Singular Value Decomposition ..............410
5.13 Orthogonal Projection ..................428
5.14 Why Least Squares? ...................445
5.15 Angles between Subspaces ................449
6. Determinants .................459
6.1 Determinants .......................459
6.2 Additional Properties of Determinants ..........475
7. Eigenvalues and Eigenvectors ..........489
7.1 Elementary Properties of Eigensystems ..........489
7.2 Diagonalization by Similarity Transformations .......505
7.3 Functions of Diagonalizable Matrices ...........525
7.4 Systems Of Differential Equations .............541
7.5 Normal Matrices .....................547
7.6 Positive Definite Matrices .................558
7.7 Nilpotent Matrices and Jordan Structure .........573
7.8 Jordan Form .......................586
7.9 Functions of Nondiagonalizable Matrices .........598
7.10 Difference Equations, Limits, and Summability ......615
7.11 Minimum Polynomials and Krylov Methods .......641
8 Perron–Frobenius Theory of Nonnegative Matrices . 661
8.1 Introduction ........................661
8.2 Positive Matrices .....................663
8.3 Nonnegative Matrices ...................670
8.4 Stochastic Matrices and Markov Chains ..........687
CHAPTER 1
Linear
Equations
1.1 INTRODUCTION
A fundamental problem that surfaces in all mathematical sciences is that of
analyzing and solving m algebraic equations in n unknowns. The study of a
system of simultaneous linear equations is in a natural and indivisible alliance
with the study of the rectangular array of numbers defined by the coefficients of
the equations. This link seems to have been made at the outset.
The earliest recorded analysis of simultaneous equations is found in the
ancient Chinese book Chiu-chang Suan-shu (Nine Chapters on Arithmetic), es-
timated to have been written some time around 200 B.C. In the beginning of
Chapter VIII, there appears a problem of the following form.
Three sheafs of a good crop, two sheafs of a mediocre crop, and
one sheaf of a bad crop are sold for 39 dou. Two sheafs of
good, three mediocre, and one bad are sold for 34 dou; and one
good, two mediocre, and three bad are sold for 26 dou. What is
the price received for each sheaf of a good crop, each sheaf of a
mediocre crop, and each sheaf of a bad crop?
Today, this problem would be formulated as three equations in three un-
knowns by writing
3x +2y + z =39,
2x +3y + z =34,
x +2y +3z =26,
where x, y, and z represent the price for one sheaf of a good, mediocre, and
bad crop, respectively. The Chinese saw right to the heart of the matter. They
placed the coefficients (represented by colored bamboo rods) of this system in
2 Chapter 1 Linear Equations
a square array on a “counting board” and then manipulated the lines of the
array according to prescribed rules of thumb. Their counting board techniques
and rules of thumb found their way to Japan and eventually appeared in Europe
with the colored rods having been replaced by numerals and the counting board
replaced by pen and paper. In Europe, the technique became known as Gaussian
elimination in honor of the German mathematician Carl Gauss,
1
whose extensive
use of it popularized the method.
Because this elimination technique is fundamental, we begin the study of
our subject by learning how to apply this method in order to compute solutions
for linear equations. After the computational aspects have been mastered, we
will turn to the more theoretical facets surrounding linear systems.
1
Carl Friedrich Gauss (1777–1855) is considered by many to have been the greatest mathemati-
cian who has ever lived, and his astounding career requires several volumes to document. He
was referred to by his peers as the “prince of mathematicians.” Upon Gauss’s death one of
them wrote that “His mind penetrated into the deepest secrets of numbers, space, and nature;
He measured the course of the stars, the form and forces of the Earth; He carried within himself
the evolution of mathematical sciences of a coming century.” History has proven this remark
to be true.
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