Contemporary linear algebra

639 Pages·2008·50.876 MB·English

by  Anton, Howard, Busby, Robert C.

CONTEMPORARY
4KλVA
Contemporary
Linear Algebra
Howard Anton
Drexel University
Robert C. Busby
Drexel University
John Wiley & Sons, Inc.
ACQUISITIONS EDITOR Laurie Rosatone
MARKETING MANAGER Julie Z. Lindstrom
SENIOR PRODUCTION EDITOR Ken Santor
PHOTO EDITOR Sara Wight
COVER DESIGN Madelyn Lesure
ILLUSTRATION STUDIO ATI Illustration Services
Cover Art: The mandrill picture on this cover has long been a favorite of researchers
studying techniques of image compression. The colored image in the center was scanned
in black and white and the bordering images were rendered with various levels of image
compression using the method of “singular value decomposition” discussed in this text.
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10 9
ABOUT THE AUTHORS
Howard Anton obtained his B.A. from Lehigh University, his Robert C. Busby obtained his B.S. in physics from Drexel
M.A. from the University of Illinois, and his Ph.D. from the University and his M.A. and Ph.D. in mathematics from the
Polytechnic University of Brooklyn, all in mathematics. In the University of Pennsylvania. He taught at Oakland University
early 1960s he worked for Burroughs Corporation and Avco in Rochester, Michigan, and since 1969 has taught full time
Corporation at Cape Canaveral, Florida (now the Kennedy at Drexel University, where he currently holds the position of
Space Center), on mathematical problems related to the Professor in the Department of Mathematics and Computer
manned space program. In 1968 he joined the Mathematics Science. He has regularly taught courses in calculus, lin­
Department at Drexel University, where he taught full time ear algebra, probability and statistics, and modem analysis.
until 1983. Since then he has been an adjunct professor at Dr. Busby is the author of numerous research articles in func­
Drexel and has devoted the majority of his time to textbook tional analysis, representation theory, and operator algebras,
writing and projects for mathematical associations. He was and he has coauthored an undergraduate text in discrete math­
president of the Eastern Pennsylvania and Delaware section ematical structures and a workbook on the use of Maple in
of the Mathematical Association of America (MAA), served calculus. His current professional interests include aspects
on the Board of Governors of the MAA, and guided the cre­ of signal processing and the use of computer technology in
ation of its Student Chapters. He has published numerous undergraduate education. Professor Busby also enjoys con­
research papers in functional analysis, approximation theory, temporaryjazz and computer graphic design. He and his wife,
and topology, as well as various pedagogical papers. He is Patricia, have two sons, Robert and Scott.
best known for his textbooks in mathematics, which have
been widely used for more than thirty years. There are cur­
rently more than 125 versions of his books used through­
out the world, including translations into Spanish, Arabic,
Portuguese, Italian, Indonesian, French, Japanese, Chinese,
Hebrew, and German. In 1994 he was awarded the Textbook
Excellence Award by the Textbook Authors Association. For
relaxation, Dr. Anton enjoys traveling, photography, and art.
CONTENTS
CHAPTER 1 Vectors 1 chapter 7 Dimension and Structure 329
1.1 Vectors and Matrices in Engineering and 7.1 Basis and Dimension 329
Mathematics; n-Space 1 7.2 Properties of Bases 335
1.2 Dot Product and Orthogonality 15 7.3 The Fundamental Spaces of a Matrix 342
1.3 Vector Equations of Lines and Planes 29 7.4 The Dimension Theorem and Its Implications 352
7.5 The Rank Theorem and Its Implications 360
chapter 2 Systems of Linear Equations 39
7.6 The Pivot Theorem and Its Implications 370
2.1 Introduction to Systems of Linear Equations 39
7.7 The Projection Theorem and Its Implications 379
2.2 Solving Linear Systems by Row Reduction 48
7.8 Best Approximation and Least Squares 393
2.3 Applications of Linear Systems 63
7.9 Orthonormal Bases and the Gram-Schmidt
chapter 3 Matrices and Matrix Algebra 79 Process 406
7.10 (//[-Decomposition; Householder
3.1 Operations on Matrices 79
Transformations 417
3.2 Inverses; Algebraic Properties of Matrices 94
7.11 Coordinates with Respect to a Basis 428
3.3 Elementary Matrices; A Method for
Finding A-1 109
chapter 8 Diagonalization 443
3.4 Subspaces and Linear Independence 123
8.1 Matrix Representations of Linear
3.5 The Geometry of Linear Systems 135
Transformations 443
3.6 Matrices with Special Forms 143
8.2 Similarity and Diagonalizability 456
3.7 Matrix Factorizations;/.[/-Decomposition 154
8.3 Orthogonal Diagonalizability; Functions
3.8 Partitioned Matrices and Parallel Processing 166 of a Matrix 468
8.4 Quadratic Forms 481
chapter 4 Determinants 175
8.5 Application of Quadratic Forms
4.1 Determinants; Cofactor Expansion 175
to Optimization 495
4.2 Properties of Determinants 184
8.6 Singular Value Decomposition 502
4.3 Cramer’s Rule; Formula for A^^l; Applications of
8.7 The Pseudoinverse 518
Determinants 196
8.8 Complex Eigenvalues and Eigenvectors 525
4.4 A First Look at Eigenvalues and Eigenvectors 210
8.9 Hermitian, Unitary, and Normal Matrices 535
chapter 5 Matrix Models 225 8.10 Systems of Differential Equations 542
5.1 Dynamical Systems and Markov Chains 225
chapter 9 General Vector Spaces 555
5.2 Leontief Input-Output Models 235
9.1 Vector Space Axioms 555
5.3 Gauss-Seidel and Jacobi Iteration; Sparse Linear
9.2 Inner Product Spaces; Fourier Series 569
Systems 241
9.3 General Linear Transformations;
5.4 The Power Method; Application to Internet Search
Isomorphism 582
Engines 249
chapter 6 Linear Transformations 265 appendix a How to Read Theorems ai
6.1 Matrices as Transformations 265
appendix b Complex Numbers A3
6.2 Geometry of Linear Operators 280
6.3 Kernel and Range 296
ANSWERS TO ODD-NUMBERED EXERCISES A9
6.4 Composition and Invertibility of Linear
Transformations 305 PHOTO CREDITS C1
6.5 Computer Graphics 318 INDEX 1-1
xi
GUIDE FOR THE INSTRUCTOR
Number of Lectures
The Syllabus Guide below provides for a 29-lecture core and a core topic, you can do so by adjusting the number of starred
a 35-lecture core. The 29-lecture core is for schools with topics that you cover.
time constraints, as with abbreviated summer courses. Both By the end of Lecture 15 the following concepts will have
core programs can be supplemented by starred topics as time been covered in a basic form: linear combination, spanning,
permits. The omission of starred topics does not affect the subspace, dimension, eigenvalues, and eigenvectors. Thus,
readability or continuity of the core topics. even with a relatively slow pace you will have no trouble
touching on all of the main ideas in the course.
Pace
The core program is based on covering one section per lec­ Organization
ture, but whether you can do this in every instance will depend It is our feeling that the most effective way to teach abstract
on your teaching style and the capabilities of your particular vector spaces is to place that material at the end (Chapter 9),
students. For longer sections we recommend that you just at which point it occurs as a “natural generalization” of the
highlight the main points in class and leave the details for the earlier material, and the student has developed the “linear al­
students to read. Since the reviews of this text have praised gebra maturity” to understand its purpose. However, we rec­
the clarity of the exposition, you should find this workable. If, ognize that not everybody shares that philosophy, so we have
in certain cases, you want to devote more than one lecture to designed that chapter so it can be moved forward, if desired.
SYLLABUS GUIDE
35-Lecture 29-Lecture
Course CONTENTS Course
Chapter 1 Vectors Chapter 1
1 1.1 Vectors and Matrices in Engineering and Mathematics; n-Space 1
2 1.2 Dot Product and Orthogonality 2
3 1.3 Vector Equations of Lines and Planes 3
Chapter 2 Systems of Linear Equations Chapter 2
4 2.1 Introduction to Systems of Linear Equations 4
5 2.2 Solving Linear Systems by Row Reduction 5
* 2.3 Applications of Linear Systems *
Chapter 3 Matrices and Matrix Algebra Chapter 3
6 3.1 Operations on Matrices 6
7 3.2 Inverses; Algebraic Properties of Matrices 7
8 3.3 Elementary Matrices; A Method for Finding A^l 8
9 3.4 Subspaces and Linear Independence 9
10 3.5 The Geometry of Linear Systems 10
11 3.6 Matrices with Special Forms 11
* 3.7 Matrix Factorizations; LU -Decomposition *
12 3.8 Partitioned Matrices and Parallel Processing 12
Chapter 4 Determinants Chapter 4
13 4.1 Determinants; Cofactor Expansion 13
14 4.2 Properties of Determinants *
xii
Guide for the Instructor xiii
*
4.3 Cramer’s Rule; Formula for A _l; Applications of Determinants *
15 4.4 A First Look at Eigenvalues and Eigenvectors 14
Chapter 5 Matrix Models Chapter 5
* 5.1 Dynamical Systems and Markov Chains *
* 5.2 Leontief Input-Output Models *
* 5.3 Gauss-Seidel and Jacobi Iteration; Sparse Linear Systems *
* 5.4 The Power Method; Application to Internet Search Engines *
Chapter 6 Linear Transformations Chapter 6
16 6.1 Matrices as Transformations 15
17 6.2 Geometry of Linear Operators 16
18 6.3 Kernel and Range 17
19 6.4 Composition and Invertibility of Linear Transformations *
* 6.5 Computer Graphics *
Chapter 7 Dimension and Structure Chapter 7
20 7.1 Basis and Dimension 18
21 7.2 Properties of Bases 19
22 7.3 The Fundamental Spaces of a Matrix 20
23 7.4 The Dimension Theorem and Its Implications 21
24 7.5 The Rank Theorem and Its Implications *
25 7.6 The Pivot Theorem and Its Implications *
26 7.7 The Projection Theorem and Its Implications 22
27 7.8 Best Approximation and Least Squares 23
28 7.9 Orthonormal Bases and the Gram-Schmidt Process *
* 7.10 (^-Decomposition; Householder Transformations *
29 7.11 Coordinates with Respect to a Basis 24
Chapter 8 Diagonalization Chapter 8
30 8.1 Matrix Representations of Linear Transformations 25
31 8.2 Similarity and Diagonalizability 26
32 8.3 Orthogonal Diagonalizability; Functions of a Matrix 27
* 8.4 Quadratic Forms *
* 8.5 Application of Quadratic Forms to Optimization *
* 8.6 Singular Value Decomposition ⅛
* 8.7 The Pseudoinverse *
* 8.8 Complex Eigenvalues and Eigenvectors *
* 8.9 Hermitian. Unitary, and Normal Matrices ⅛
* 8.10 Systems of Differential Equations *
Chapter 9 General Vector Spaces Chapter 9
33 9.1 Vector Space Axioms 28
34 9.2 Inner Product Spaces; Fourier Series 29
35 9.3 General Linear Transformations; Isomorphism *
Appendices
Appendix A How to Read Theorems
Appendix B Complex Numbers
TOPIC PLANNER
To assist you in planning your course, we have provided below a list of topics that occur in each section. These topics are
identified in the text by headings in the margin. You will find additional lecture planning information on the Web site for this text.
CHAPTER 1 VECTORS and Columns of a Matrix Product ∙ Matrix Products as Lin­
ear Combinations ∙ Transpose of a Matrix ∙ Trace ∙ Inner
1.1 Vectors and Matrices in Engineering and Mathe­
and Outer Matrix Products
matics; n-Space Scalars and Vectors ∙ Equivalent Vectors
• Vector Addition ∙ Vector Subtraction ∙ Scalar Multipli­ 3.2 Inverses; Algebraic Properties of Matrices Properties
cation ∙ Vectors in Coordinate Systems ∙ Components of of Matrix Addition and Scalar Multiplication ∙ Properties of
a Vector Whose Initial Point Is Not at the Origin ∙ Matrix Multiplication ∙ Zero Matrices ∙ Identity Matrices ∙
Vectors in R" ∙ Equality of Vectors ∙ Sums of Three Inverse of a Matrix ∙ Properties of Inverses ∙ Powers of a
or More Vectors ∙ Parallel and Collinear Vectors ∙ Linear Matrix ∙ Matrix Polynomials ∙ Properties of the Transpose
Combinations ∙ Application to Computer Color Models ∙ • Properties of the Trace ∙ Transpose and Dot Product
Alternative Notations for Vectors ∙ Matrices
3.3 Elementary Matrices; A Method for Finding A-1
1.2 Dot Product and Orthogonality Norm of a Vector Elementary Matrices ∙ Characterizations of Invertibility ∙
• Unit Vectors ∙ The Standard Unit Vectors ∙ Distance Row Equivalence ∙ An Algorithm for Inverting Matrices
Between Points in R" ∙ Dot Products ∙ Algebraic Properties • Solving Linear Systems by Matrix Inversion ∙ Solving
of the Dot Product ∙ Angle Between Vectors in R2 and R3 Multiple Linear Systems with a Common Coefficient Matrix
• Orthogonality ∙ Orthonormal Sets ∙ Euclidean Geometry • Consistency of Linear Systems
in Rn
3.4 Subspaces and Linear Independence Subspaces of
1.3 Vector Equations of Lines and Planes Vector and Rn ∙ Solution Space of a Linear System ∙ Linear Inde­
Parametric Equations of Lines ∙ Lines Through Two Points pendence ∙ Linear Independence and Homogeneous Linear
• Point-Normal Equations of Planes ∙ Vector and Parametric Systems ∙ Translated Subspaces ∙ A Unifying Theorem
Equations of Planes ∙ Lines and Planes in R" ∙ Comments
3.5 The Geometry of Linear Systems The Relationship
on Terminology
Between Ax = b and Ax = 0 ∙ Consistency of a Linear
System from the Vector Point of View ∙ Hyperplanes ∙
CHAPTER 2 SYSTEMS OF LINEAR EQUATIONS
Geometric Interpretations of Solution Spaces
2.1 Introduction to Systems of Linear Equations Linear
3.6 Matrices with Special Forms Diagonal Matrices
Systems ∙ Linear Systems with Two and Three Unknowns ∙
• Triangular Matrices ∙ Linear Systems with Triangular
Augmented Matrices and Elementary Row Operations
Coefficient Matrices ∙ Properties of Triangular Matrices ∙
2.2 Solving Linear Systems by Row Reduction Consider­ Symmetric and Skew-Symmetric Matrices ∙ Invertibility of
ations in Solving Linear Systems ∙ Echelon Forms ∙ General Symmetric Matrices ∙ Matrices of the Form AτA and AAr ∙
Solutions as Linear Combinations of Column Vectors ∙ Fixed Points of a Matrix ∙ A Technique for Inverting I — A
Gauss-Jordan and Gaussian Elimination ∙ Some Facts About When A Is Nilpotent ∙ Inverting I — A by Power Series
Echelon Forms ∙ Back Substitution ∙ Homogeneous Linear
3.7 Matrix Factorizations; LL7-Dccomposition Solving
Systems ∙ The Dimension Theorem for Homogeneous Linear
Linear Systems by Factorization ∙ Finding LL∕-Decom­
Systems ∙ Stability, Roundoff Error, and Partial Pivoting
positions ∙ The Relationship Between Gaussian Elimination
2.3 Applications of Linear Systems Global Positioning ∙ and ΔL∕-Decomposition ∙ Matrix Inversion by LU-Decom­
Network Analysis ∙ Electrical Circuits ∙ Balancing Chemi­ position ∙ LDU-Decompositions ∙ Using Permutation Ma­
cal Equations ∙ Polynomial Interpolation trices to Deal with Row Interchanges ∙ Flops and the Cost of
Solving a Linear System ∙ Cost Estimates for Solving Large
CHAPTER 3 MATRICES AND MATRIX ALGEBRA Linear Systems ∙ Considerations in Choosing an Algorithm
for Solving a Linear System
3.1 Operations on Matrices Matrix Notation and Ter­
minology ∙ Operations on Matrices ∙ Row and Column 3.8 Partitioned Matrices and Parallel Processing Gen­
Vectors ∙ The Product A× ∙ The Product AB ∙ Finding eral Partitioning ∙ Block Diagonal Matrices ∙ Block Upper
Specific Entries in a Matrix Product ∙ Finding Specific Rows Triangular Matrices
xiv
Topic Planner xv
CHAPTER 4 DETERMINANTS Properties of Linear Transformations ∙ All Linear Trans­
formations from R" to R"' Are Matrix Transformations
4.1 Determinants; Cofactor Expansion Determinants of
• Rotations About the Origin ∙ Reflections About Lines
2x2 and 3 × 3 Matrices ∙ Elementary Products ∙ General
Through the Origin ∙ Orthogonal Projections onto Lines
Determinants ∙ Evaluation Difficulties for Higher-Order
Through the Origin ∙ Transformations of the Unit Square ∙
Determinants ∙ Determinants of Matrices with Rows or
Power Sequences
Columns That Have All Zeros ∙ Determinants of Triangular
Matrices ∙ Minors and Cofactors ∙ Cofactor Expansions
6.2 Geometry of Linear Operators Norm-Preserving
4.2 Properties of Determinants Determinant of Aτ Linear Operators ∙ Orthogonal Operators Preserve Angles
• Effect of Elementary Row Operations on a Determi­ and Orthogonality ∙ Orthogonal Matrices ∙ All Orthogonal
nant ∙ Simplifying Cofactor Expansions ∙ Determinants by Linear Operators on R2 are Rotations or Reflections ∙
Gaussian Elimination ∙ A Determinant Test for Invertibility Contractions and Dilations of R2 ∙ Vertical and Horizontal
• Determinant of a Product of Matrices ∙ Determinant Compressions and Expansions of R2 ∙ Shears ∙ Linear
Evaluation by LU -Decomposition ∙ Determinant of the Operators on R3 ∙ Reflections About Coordinate Planes ∙
Inverse of a Matrix ∙ Determinant of A + B ∙ A Unifying Rotations in R3 ∙ General Rotations
Theorem
6.3 Kernel and Range Kernel of a Linear Transformation
4.3 Cramer’s Rule; Formula for A-1; Applications
• Kernel of a Matrix Transformation ∙ Range of a Linear
of Determinants Adjoint of a Matrix ∙ A Formula for the
Transformation ∙ Range of a Matrix Transformation ∙
Inverse of a Matrix ∙ How the Inverse Formula Is Actually
Existence and Uniqueness Issues ∙ One-to-One and Onto
Used ∙ Cramer’s Rule ∙ Geometric Interpretation of Deter­ from the Viewpoint of Linear Systems ∙ A Unifying Theorem
minants ∙ Polynomial Interpolation and the Vandermonde
Determinant ∙ Cross Products 6.4 Composition and Invertibility of Linear Transfor­
mations Compositions of Linear Transformations ∙
4.4 A First Look at Eigenvalues and Eigenvectors Fixed
Compositions of Three or More Linear Transformations
Points ∙ Eigenvalues and Eigenvectors ∙ Eigenvalues of
• Factoring Linear Operators into Compositions ∙ Inverse
Triangular Matrices ∙ Eigenvalues of Powers of a Matrix
of a Linear Transformation ∙ Invertible Linear Operators ∙
• A Unifying Theorem ∙ Complex Eigenvalues ∙ Algebraic
Geometric Properties of Invertible Linear Operators on R2 ∙
Multiplicity ∙ Eigenvalue Analysis of 2x2 Matrices ∙
Image of the Unit Square Under an Invertible Linear Operator
Eigenvalue Analysis of 2 × 2 Symmetric Matrices ∙ Expres­
sions for Determinant and Trace in Terms of Eigenvalues ∙ 6.5 Computer Graphics Wireframes ∙ Matrix Represen­
Eigenvalues by Numerical Methods tations of Wireframes ∙ Transforming Wireframes ∙ Transla­
tion Using Homogeneous Coordinates ∙ Three-Dimensional
CHAPTERS MATRIX MODELS Graphics
5.1 Dynamical Systems and Markov Chains Dynamical
Systems ∙ Markov Chains ∙ Markov Chains as Powers of the CHAPTER 7 DIMENSION AND STRUCTURE
Transition Matrix ∙ Long-Term Behavior of a Markov Chain
7.1 Basis and Dimension Bases for Subspaces ∙ Dimen­
5.2 Leontief Input-Output Models Inputs and Outputs in sion of a Solution Space ∙ Dimension of a Hyperplane
an Economy ∙ The Leontief Model of an Open Economy ∙
Productive Open Economies 7.2 Properties of Bases Properties of Bases ∙ Subspaces
of Subspaces ∙ Sometimes Spanning Implies Linear Indepen­
5.3 Gauss-Seidel and Jacobi Iteration; Sparse Linear
dence and Conversely ∙ A Unifying Theorem
Systems Iterative Methods ∙ Jacobi Iteration ∙ Gauss-
Seidel Iteration ∙ Convergence ∙ Speeding Up Convergence 7.3 The Fundamental Spaces of a Matrix The Fun­
damental Spaces of a Matrix ∙ Orthogonal Complements ∙
5.4 The Power Method; Application to Internet Search
Properties of Orthogonal Complements ∙ Finding Bases by
Engines The Power Method ∙ The Power Method with
Row Reduction ∙ Determining Whether a Vector Is in a Given
Euclidean Scaling ∙ The Power Method with Maximum
Subspace
Entry Scaling ∙ Rate of Convergence ∙ Stopping Procedures
• An Application of the Power Method to Internet Searches ∙
7.4 The Dimension Theorem and Its Implications The
Variations of the Power Method
Dimension Theorem for Matrices ∙ Extending a Linearly
Independent Set to a Basis ∙ Some Consequences of the
CHAPTER 6 LINEAR TRANSFORMATIONS
Dimension Theorem for Matrices ∙ The Dimension Theorem
6.1 Matrices as Transformations A Review of Functions for Subspaces ∙ A Unifying Theorem ∙ More on Hyper­
• Matrix Transformations ∙ Linear Transformations ∙ Some planes ∙ Rank 1 Matrices ∙ Symmetric Rank 1 Matrices
xvi Topic Planner
7.5 The Rank Theorem and Its Implications The Rank 8.2 Similarity and Diagonalizability Similar Matrices
Theorem ∙ Relationship Between Consistency and Rank • Similarity Invariants ∙ Eigenvectors and Eigenvalues of
• Overdetermined and Underdeteπnined Linear Systems ∙ Similar Matrices ∙ Diagonalization ∙ A Method for Diago­
Matrices of the form ArA and AAl ∙ Some Unifying nalizing a Matrix ∙ Linear Independence of Eigenvectors ∙
Theorems ∙ Applications of Rank Relationship Between Algebraic and Geometric Multiplicity
• A Unifying Theorem on Diagonalizability
7.6 The Pivot Theorem and Its Implications Basis
Problems Revisited ∙ Bases for the Fundamental Spaces
8.3 Orthogonal Diagonalizability; Functions of a Matrix
of a Matrix ∙ A Column-Row Factorization ∙ Column-Row
Orthogonal Similarity ∙ A Method for Orthogonally Di­
Expansion
agonalizing a Symmetric Matrix ∙ Spectral Decomposition
7.7 The Projection Theorem and Its Implications Or­ • Powers of a Diagonalizable Matrix ∙ Cayley-Hamilton
thogonal Projections onto Lines Through the Origin in R2 Theorem ∙ Exponential of a Matrix ∙ Diagonalization and
• Orthogonal Projections onto Lines Through the Origin in Linear Systems ∙ The Nondiagonalizable Case
R" ∙ Projection Operators on Rn ∙ Orthogonal Projections
8.4 Quadratic Forms Definition of a Quadratic Form
onto General Subspaces ∙ When Does a Matrix Represent
• Change of Variable in a Quadratic Form ∙ Quadratic
an Orthogonal Projection? ∙ Strang Diagrams ∙ Full Column
Forms in Geometry ∙ Identifying Conic Sections ∙ Positive
Rank and Consistency of a Linear System ∙ The Double Perp
Definite Quadratic Forms ∙ Classifying Conic Sections
Theorem ∙ Orthogonal Projections onto Wj-
Using Eigenvalues ∙ Identifying Positive Definite Matrices
7.8 Best Approximation and Least Squares Minimum
• Cholesky Factorization
Distance Problems ∙ Least Squares Solutions of Linear
Systems ∙ Finding Least Squares Solutions of Linear 8.5 Application of Quadratic Forms to Optimization
Systems ∙ Orthogonality Property of Least Squares Error Relative Extrema of Functions of Two Variables ∙ Con­
Vectors ∙ Strang Diagrams for Least Squares Problems ∙ strained Extremum Problems ∙ Constrained Extrema and
Fitting a Curve to Experimental Data ∙ Least Squares Fits Level Curves
by Higher-Degree Polynomials ∙ Theory Versus Practice
8.6 Singular Value Decomposition Singular Value De­
7.9 Orthonormal Bases and the Gram-Schmidt Process
composition of Square Matrices ∙ Singular Value Decom­
Orthogonal and Orthonormal Bases ∙ Orthogonal Projections
position of Symmetric Matrices ∙ Polar Decomposition ∙
Using Orthogonal Bases ∙ Trace and Orthogonal Projections
Singular Value Decomposition of Nonsquare Matrices ∙
• Linear Combinations of Orthonormal Basis Vectors ∙
Singular Value Decomposition and the Fundamental Spaces
Finding Orthogonal and Orthonormal Bases ∙ A Property of
of a Matrix ∙ Reduced Singular Value Decomposition ∙
the Gram-Schmidt Process ∙ Extending Orthonormal Sets to
Data Compression and Image Processing ∙ Singular Value
Orthonormal Bases
Decomposition from the Transformation Point of View
7.10 QR-Decomposition; Householder Transformations
8.7 The Pseudoinverse The Pseudoinverse ∙ Properties of
(^-Decomposition ∙ The Role of βR-Decomposition in
the Pseudoinverse ∙ The Pscudoinverse and Least Squares ∙
Least Squares Problems ∙ Other Numerical Issues ∙ House­
Condition Number and Numerical Considerations
holder Reflections ∙ (^-Decomposition Using Householder
Reflections ∙ Householder Reflections in Applications
8.8 Complex Eigenvalues and Eigenvectors Vectors in
7.11 Coordinates with Respect to a Basis Nonrectangular Cn ∙ Algebraic Properties of the Complex Conjugate ∙ The
Coordinate Systems in R2 and Ri ∙ Coordinates with Respect Complex Euclidean Inner Product ∙ Vector Space Concepts
to an Orthonormal Basis ∙ Computing with Coordinates with in Cn ∙ Complex Eigenvalues of Real Matrices Acting on
Respect to an Orthononormal Basis ∙ Change of Basis for R', Vectors in C" ∙ A Proof That Real Symmetric Matrices Have
• Invertibility of Transition Matrices ∙ A Good Technique for Real Eigenvalues ∙ A Geometric Interpretation of Complex
finding Transition Matrices ∙ Coordinate Maps ∙ Transition Eigenvalues of Real Matrices
Between Orthonormal Bases ∙ Application to Rotation of
8.9 Hermitian, Unitary, and Normal Matrices Hermitian
Coordinate Axes ∙ New Ways to Think About Matrices
and Unitary Matrices ∙ Unitary Diagonalizability ∙ Skew-
Hermitian Matrices ∙ Normal Matrices ∙ A Comparison of
CHAPTER 8 DIAGONALIZATION
Eigenvalues
8.1 Matrix Representations of Linear Transformations
Matrix of a Linear Operator with Respect to a Basis ∙ 8.10 Systems of Differential Equations Terminology
Changing Bases ∙ Matrix of a Linear Transformation with • Linear Systems of Differential Equations ∙ Fundamental
Respect to a Pair of Bases ∙ Effect of Changing Bases on Solutions ∙ Solutions Using Eigenvalues and Eigenvectors ∙
Matrices of Linear Transformations ∙ Representing Linear Exponential Form of a Solution ∙ The Case Where A is Not
Operators with Two Bases Diagonalizable