Advanced 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