Matrix Analysis

Citation

Horn, R. A., & Johnson, C. R. (2013). Matrix Analysis (2nd ed.). Cambridge University Press.

Chapter Summary

The second edition has nine numbered chapters (0 to 8) and six appendices, followed by references, notation, hints for problems and an index. The first edition's preface describes the aim as a useful modern treatment of a broad range of topics that can serve as an undergraduate or graduate text and as a self-contained reference. Chapter 0 is a reference summary with no exercises. Chapters 1 to 3 contain the core material for a second course in linear algebra, and the remaining chapters are each developed substantially independently as a major topic.

Chapter 0: Review and Miscellanea

A summary of concepts and facts that serve as an as-needed reference: vector spaces, matrices, determinants (twice), rank, nonsingularity, the Euclidean inner product and norm, partitioned matrices, special types of matrices, change of basis and equivalence relations. The second edition expanded it by about 75 percent. It has no problems, and courses and self-study typically begin with Chapter 1.

Chapter 1: Eigenvalues, Eigenvectors, and Similarity

The eigenvalue–eigenvector equation, the characteristic polynomial and algebraic multiplicity, similarity, and left and right eigenvectors with geometric multiplicity. The new edition emphasizes the role of left eigenvectors.

Chapter 2: Unitary Similarity and Unitary Equivalence

Unitary matrices and the QR factorization, unitary similarity, unitary and real orthogonal triangularizations, consequences of Schur's triangularization theorem, normal matrices, unitary equivalence with the singular value decomposition, and the CS decomposition. The singular value decomposition is introduced here so that it can be used throughout the book.

Chapter 3: Canonical Forms for Similarity and Triangular Factorizations

The Jordan canonical form theorem and its consequences, the minimal polynomial and companion matrix, the real Jordan and Weyr canonical forms, and triangular factorizations. The preface says the Jordan form is approached via the Weyr characteristic.

Chapter 4: Hermitian Matrices, Symmetric Matrices, and Congruences

Properties and characterizations of Hermitian matrices, variational characterizations through subspace intersections, eigenvalue inequalities, unitary congruence and complex symmetric matrices, congruences and diagonalizations, and consimilarity and condiagonalization.

Chapter 5: Norms for Vectors and Matrices

Definitions and examples of norms and inner products, their algebraic and analytic properties, duality and geometric properties, matrix norms, vector norms on matrices, and condition numbers for inverses and linear systems.

Chapter 6: Location and Perturbation of Eigenvalues

Geršgorin discs and a closer look at them, eigenvalue perturbation theorems and other eigenvalue inclusion sets.

Chapter 7: Positive Definite and Semidefinite Matrices

Definitions, characterizations, the polar and singular value decompositions and their consequences, the Schur product theorem, simultaneous diagonalizations and convexity, the Loewner partial order and block matrices, and inequalities for positive definite matrices.

Chapter 8: Positive and Nonnegative Matrices

Inequalities and generalities, positive matrices, nonnegative matrices, irreducible and primitive matrices, a general limit theorem, and stochastic and doubly stochastic matrices. This is the Perron–Frobenius theory.

Appendices A–F

Complex numbers; convex sets and functions; the fundamental theorem of algebra; continuity of polynomial zeroes and matrix eigenvalues; continuity, compactness and Weierstrass's theorem; and canonical pairs, a table of canonical forms for pairs of Hermitian matrices or a symmetric–skew symmetric pair.

Key Concepts

Chapters 1–3: Similarity and Canonical Forms

• Schur triangularization: every complex square matrix is unitarily similar to an upper triangular matrix, with many consequences developed in Chapter 2.
• Jordan and Weyr canonical forms: the canonical forms for similarity, which the second edition uses as a unifying theme.
• Singular value decomposition: the factorization introduced in Section 2.6 and used throughout.

Chapters 4–6: Hermitian Matrices, Norms, Eigenvalue Location

• Variational characterization: describing eigenvalues of Hermitian matrices as extreme values, via subspace intersections.
• Matrix norms: norms on matrices, with the submultiplicativity axiom as a distinction in the results.
• Geršgorin discs: regions of the complex plane that contain the eigenvalues of a matrix.

Chapters 7–8: Positive and Nonnegative Matrices

• Loewner partial order: ordering Hermitian matrices by positive semidefiniteness of their difference.
• Schur product theorem: the Hadamard product of positive semidefinite matrices is positive semidefinite.
• Perron–Frobenius theory: the spectral properties of positive, nonnegative, primitive and irreducible matrices.

Critical Analysis

Strengths

• Canonical forms as organizing idea. The publisher and the second-edition preface describe canonical forms as a unifying theme across similarity, unitary equivalence, congruence and other equivalence relations.
• Large problem collection. The second edition has more than 1,100 problems, up from about 690, with hints for many in an appendix and problem threads that develop topics such as the adjugate and compound matrices, finite-dimensional quantum systems and the Loewner ellipsoid across chapters.
• Built for reference as well as study. The index grew from about 1,200 to more than 3,500 entries, and the authors say that an unfamiliar term should be looked up there, where a pointer to its definition is likely to be found.
• Modular design. The first-edition preface says that after Chapters 1 to 3 each chapter treats a major topic substantially independently, so readers can enter by interest.

Limitations

• Selection of topics is broad but not exhaustive. The first-edition preface refers to a companion volume, Topics in Matrix Analysis, for the field of values, inertia, M-matrices, matrix equations and Kronecker and Hadamard products.
• Not an introductory text. The preface assumes a one-semester elementary linear algebra course and some rudimentary analysis, and Chapters 1 to 3 are described as the core of a second course.
• A selection is expected. The authors say the book supports one- or two-semester courses through selection of chapters and sections, and recommend a careful preselection for a given course, so it is not meant to be read straight through in a single course.

Where It Fits in the Course

This is the course's advanced reference on matrix theory. Hefferon's free text, Strang's Introduction to Linear Algebra and Anton's Elementary Linear Algebra are the first courses that supply the one-semester background the authors assume. Axler's Linear Algebra Done Right is a proof-based treatment of vector spaces and operators, and Anthony and co-authors' Linear Algebra: Concepts and Methods develops core theory and methods; either works as the immediate step before Horn and Johnson.

Horn and Johnson goes beyond them in covering Schur triangularization, the Jordan form, singular values, matrix norms and Perron–Frobenius theory with the precision of research use. It is the book to keep after the course, for readers in statistics, numerical computation, control or applied analysis.

How to Use This Book

The book suits students who have completed a first linear algebra course and are comfortable with proofs and basic analysis, as well as researchers who need a reference.

• Core: Chapters 1 to 3, with Chapter 0 consulted as needed. These match a second course in linear algebra, and the authors suggest including much of Chapters 2 and 3.
• Next: facts about Hermitian matrices from Chapter 4 and about norms from Chapter 5, as the authors recommend.
• By interest: Chapter 6 for eigenvalue localization, Chapter 7 for positive definite matrices and inequalities, Chapter 8 for Perron–Frobenius theory. Appendices A to E support the proofs; Appendix F is a specialized table.

Work the problems in threads, use the hints only after attempting them, and use the index to find definitions.