Advanced Linear and Matrix Algebra

This textbook emphasizes the interplay between algebra and geometry to motivate the study of advanced linear algebra techniques. Matrices and linear transformations are presented as two sides of the same coin, with their connection motivating inquiry throughout the book. Building on a first course in linear algebra, this book offers readers a deeper understanding of abstract st...

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This textbook emphasizes the interplay between algebra and geometry to motivate the study of advanced linear algebra techniques. Matrices and linear transformations are presented as two sides of the same coin, with their connection motivating inquiry throughout the book. Building on a first course in linear algebra, this book offers readers a deeper understanding of abstract structures, matrix decompositions, multilinearity, and tensors. Concepts draw on concrete examples throughout, offering accessible pathways to advanced techniques.

Beginning with a study of vector spaces that includes coordinates, isomorphisms, orthogonality, and projections, the book goes on to focus on matrix decompositions. Numerous decompositions are explored, including the Shur, spectral, singular value, and Jordan decompositions. In each case, the author ties the new technique back to familiar ones, to create a coherent set of tools. Tensors and multilinearity complete the book, with a study of the Kronecker product, multilinear transformations, and tensor products. Throughout, “Extra Topic” sections augment the core content with a wide range of ideas and applications, from the QR and Cholesky decompositions, to matrix-valued linear maps and semidefinite programming. Exercises of all levels accompany each section.

Advanced Linear and Matrix Algebra offers students of mathematics, data analysis, and beyond the essential tools and concepts needed for further study. The engaging color presentation and frequent marginal notes showcase the author’s visual approach. A first course in proof-based linear algebra is assumed. An ideal preparation can be found in the author’s companion volume, Introduction to Linear and Matrix Algebra.

Preface
The Purpose of this Book
Continuation of Introduction to Linear and Matrix Algebra
Features of this Book
Notes in the Margin
Exercises
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Preface
The Purpose of this Book
Continuation of Introduction to Linear and Matrix Algebra
Features of this Book
Notes in the Margin
Exercises
To the Instructor and Independent Reader
Sectioning
Extra Topic Sections
Acknowledgments
Table of Contents
1 Vector Spaces
1.1 Vector Spaces and Subspaces
1.1.1 Subspaces
1.1.2 Spans, Linear Combinations, and Independence
1.1.3 Bases
Exercises
1.2 Coordinates and Linear Transformations
1.2.1 Dimension and Coordinate Vectors
1.2.2 Change of Basis
1.2.3 Linear Transformations
1.2.4 Properties of Linear Transformations
Exercises
1.3 Isomorphisms and Linear Forms
1.3.1 Isomorphisms
1.3.2 Linear Forms
1.3.3 Bilinearity and Beyond
1.3.4 Inner Products
Exercises
1.4 Orthogonality and Adjoints
1.4.1 Orthonormal Bases
1.4.2 Adjoint Transformations
1.4.3 Unitary Matrices
1.4.4 Projections
Exercises
1.5 Summary and Review
Exercises
1.A Extra Topic: More About the Trace
1.A.1 Algebraic Characterizations of the Trace
1.A.2 Geometric Interpretation of the Trace
Exercises
1.B Extra Topic: Direct Sum, Orthogonal Complement
1.B.1 The Internal Direct Sum
1.B.2 The Orthogonal Complement
1.B.3 The External Direct Sum
Exercises
1.C Extra Topic: The QR Decomposition
1.C.1 Statement and Examples
1.C.2 Consequences and Applications
Exercises
1.D Extra Topic: Norms and Isometries
1.D.1 The p-Norms
1.D.2 From Norms Back to Inner Products
1.D.3 Isometries
Exercises
2 Matrix Decompositions
2.1 The Schur and Spectral Decompositions
2.1.1 Schur Triangularization
2.1.2 Normal Matrices and the Complex Spectral Decomposition
2.1.3 The Real Spectral Decomposition
Exercises
2.2 Positive Semidefiniteness
2.2.1 Characterizing Positive (Semi)Definite Matrices
2.2.2 Diagonal Dominance and Gershgorin Discs
2.2.3 Unitary Freedom of PSD Decompositions
Exercises
2.3 The Singular Value Decomposition
2.3.1 Geometric Interpretation and the Fundamental Subspaces
2.3.2 Relationship with Other Matrix Decompositions
2.3.3 The Operator Norm
Exercises
2.4 The Jordan Decomposition
2.4.1 Uniqueness and Similarity
2.4.2 Existence and Computation
2.4.3 Matrix Functions
Exercises
2.5 Summary and Review
Exercises
2.A Extra Topic: Quadratic Forms and Conic Sections
2.A.1 Definiteness, Ellipsoids, and Paraboloids
2.A.2 Indefiniteness and Hyperboloids
Exercises
2.B Extra Topic: Schur Complements and Cholesky
2.B.1 The Schur Complement
2.B.2 The Cholesky Decomposition
Exercises
2.C Extra Topic: Applications of the SVD
2.C.1 The Pseudoinverse and Least Squares
2.C.2 Low-Rank Approximation
Exercises
2.D Extra Topic: Continuity and Matrix Analysis
2.D.1 Dense Sets of Matrices
2.D.2 Continuity of Matrix Functions
2.D.3 Working with Non-Invertible Matrices
2.D.4 Working with Non-Diagonalizable Matrices
Exercises
3 Tensors and Multilinearity
3.1 The Kronecker Product
3.1.1 Definition and Basic Properties
3.1.2 Vectorization and the Swap Matrix
3.1.3 The Symmetric and Antisymmetric Subspaces
Exercises
3.2 Multilinear Transformations
3.2.1 Definition and Basic Examples
3.2.2 Arrays
3.2.3 Properties of Multilinear Transformations
Exercises
3.3 The Tensor Product
3.3.1 Motivation and Definition
3.3.2 Existence and Uniqueness
3.3.3 Tensor Rank
Exercises
3.4 Summary and Review
Exercises
3.A Extra Topic: Matrix-Valued Linear Maps
3.A.1 Representations
3.A.2 The Kronecker Product of Matrix-Valued Maps
3.A.3 Positive and Completely Positive Maps
Exercises
3.B Extra Topic: Homogeneous Polynomials
3.B.1 Powers of Linear Forms
3.B.2 Positive Semidefiniteness and Sums of Squares
3.B.3 Biquadratic Forms
Exercises
3.C Extra Topic: Semidefinite Programming
3.C.1 The Form of a Semidefinite Program
3.C.2 Geometric Interpretation and Solving
3.C.3 Duality
Exercises
Appendix A: Mathematical Preliminaries
A.1 Review of Introductory Linear Algebra
A.1.1 Systems of Linear Equations
A.1.2 Matrices as Linear Transformations
A.1.3 The Inverse of a Matrix
A.1.4 Range, Rank, Null Space, and Nullity
A.1.5 Determinants and Permutations
A.1.6 Eigenvalues and Eigenvectors
A.1.7 Diagonalization
A.2 Polynomials and Beyond
A.2.1 Monomials, Binomials and Multinomials
A.2.2 Taylor Polynomials and Taylor Series
A.3 Complex Numbers
A.3.1 Basic Arithmetic and Geometry
A.3.2 The Complex Conjugate
A.3.3 Euler's Formula and Polar Form
A.4 Fields
A.5 Convexity
A.5.1 Convex Sets
A.5.2 Convex Functions
Appendix B: Additional Proofs
B.1 Equivalence of Norms
B.2 Details of the Jordan Decomposition
B.3 Strong Duality for Semidefinite Programs
Appendix C: Selected Exercise Solutions
Bibliography
Index
Symbol Index
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