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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 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

Nathaniel Johnston is an Associate Professor of Mathematics at Mount Allison University in New Brunswick, Canada. His research makes use of linear algebra, matrix analysis, and convex optimization to tackle questions related to the theory of quantum entanglement. His companion volume, Introduction to Linear and Matrix Algebra, is also published by Springer.

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