Matrix Analysis and Applied Linear Algebra.pdf

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Contents
Preface
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1.
1.1
1.2
1.3
1.4
1.5
1.6
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ix
Linear Equations . . . . . . . . . . . . . . 1
Introduction . . . . . . . . . . .
Gaussian Elimination and Matrices .
Gauss–Jordan Method . . . . . . .
Two-Point Boundary Value Problems
Making Gaussian Elimination Work .
Ill-Conditioned Systems . . . . . .
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1
3
15
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33
2.
Rectangular Systems and Echelon Forms . . .
2.1
2.2
2.3
2.4
2.5
2.6
Row Echelon Form and Rank .
Reduced Row Echelon Form .
Consistency of Linear Systems
Homogeneous Systems . . . .
Nonhomogeneous Systems . .
Electrical Circuits . . . . . .
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41
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3.
Matrix Algebra . . . . . . . . . . . . . .
3.1
3.2
3.3
3.4
3.5
3.6
3.7
3.8
3.9
3.10
From Ancient China to Arthur Cayley
Addition and Transposition . . . .
Linearity . . . . . . . . . . . . .
Why Do It This Way . . . . . . .
Matrix Multiplication . . . . . . .
Properties of Matrix Multiplication .
Matrix Inversion . . . . . . . . .
Inverses of Sums and Sensitivity . .
Elementary Matrices and Equivalence
The LU Factorization . . . . . . .
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141
4.
Vector Spaces . . . . . . . . . . . . . . . 159
4.1
4.2
4.3
4.4
Spaces and Subspaces . . .
Four Fundamental Subspaces
Linear Independence . . .
Basis and Dimension . . .
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159
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vi
Contents
4.5
4.6
4.7
4.8
4.9
More about Rank . . . . . .
Classical Least Squares . . .
Linear Transformations . . .
Change of Basis and Similarity
Invariant Subspaces . . . . .
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210
223
238
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5.
Norms, Inner Products, and Orthogonality
5.1
5.2
5.3
5.4
5.5
5.6
5.7
5.8
5.9
5.10
5.11
5.12
5.13
5.14
5.15
Vector Norms . . . . . . . .
Matrix Norms . . . . . . . .
Inner-Product Spaces . . . . .
Orthogonal Vectors . . . . . .
Gram–Schmidt Procedure . . .
Unitary and Orthogonal Matrices
Orthogonal Reduction . . . . .
Discrete Fourier Transform . . .
Complementary Subspaces . . .
Range-Nullspace Decomposition
Orthogonal Decomposition . . .
Singular Value Decomposition .
Orthogonal Projection . . . . .
Why Least Squares? . . . . . .
Angles between Subspaces . . .
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269
279
286
294
307
320
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356
383
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446
450
6.
Determinants . . . . . . . . . . . . . . . 459
6.1
6.2
Determinants . . . . . . . . . . . . . . . . .
Additional Properties of Determinants . . . . . .
459
475
7.
Eigenvalues and Eigenvectors . . . . . . . . 489
7.1
7.2
7.3
7.4
7.5
7.6
7.7
7.8
7.9
Elementary Properties of Eigensystems . . .
Diagonalization by Similarity Transformations
Functions of Diagonalizable Matrices . . . .
Systems of Differential Equations . . . . . .
Normal Matrices . . . . . . . . . . . . .
Positive Definite Matrices . . . . . . . . .
Nilpotent Matrices and Jordan Structure . .
Jordan Form . . . . . . . . . . . . . . .
Functions of Nondiagonalizable Matrices . . .
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489
505
525
541
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558
574
587
599
Contents
vii
7.10
7.11
Difference Equations, Limits, and Summability . .
Minimum Polynomials and Krylov Methods . . .
616
642
8.
Perron–Frobenius Theory
8.1
8.2
8.3
8.4
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661
663
670
687
705
Introduction . . . . . . . . . . . . . . . . .
Positive Matrices . . . . . . . . . . . . . . .
Nonnegative Matrices . . . . . . . . . . . . .
Stochastic Matrices and Markov Chains . . . . .
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Index
Preface
Scaffolding
Reacting to criticism concerning the lack of motivation in his writings,
Gauss remarked that architects of great cathedrals do not obscure the beauty
of their work by leaving the scaffolding in place after the construction has been
completed. His philosophy epitomized the formal presentation and teaching of
mathematics throughout the nineteenth and twentieth centuries, and it is still
commonly found in mid-to-upper-level mathematics textbooks. The inherent ef-
ficiency and natural beauty of mathematics are compromised by straying too far
from Gauss’s viewpoint. But, as with most things in life, appreciation is gen-
erally preceded by some understanding seasoned with a bit of maturity, and in
mathematics this comes from seeing some of the scaffolding.
Purpose, Gap, and Challenge
The purpose of this text is to present the contemporary theory and applica-
tions of linear algebra to university students studying mathematics, engineering,
or applied science at the postcalculus level. Because linear algebra is usually en-
countered between basic problem solving courses such as calculus or differential
equations and more advanced courses that require students to cope with mathe-
matical rigors, the challenge in teaching applied linear algebra is to expose some
of the scaffolding while conditioning students to appreciate the utility and beauty
of the subject. Effectively meeting this challenge and bridging the inherent gaps
between basic and more advanced mathematics are primary goals of this book.
Rigor and Formalism
To reveal portions of the scaffolding, narratives, examples, and summaries
are used in place of the formal definition–theorem–proof development. But while
well-chosen examples can be more effective in promoting understanding than
rigorous proofs, and while precious classroom minutes cannot be squandered on
theoretical details, I believe that all scientifically oriented students should be
exposed to some degree of mathematical thought, logic, and rigor. And if logic
and rigor are to reside anywhere, they have to be in the textbook. So even when
logic and rigor are not the primary thrust, they are always available. Formal
definition–theorem–proof designations are not used, but definitions, theorems,
and proofs nevertheless exist, and they become evident as a student’s maturity
increases. A significant effort is made to present a linear development that avoids
forward references, circular arguments, and dependence on prior knowledge of the
subject. This results in some inefficiencies—e.g., the matrix 2-norm is presented

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