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Matrix analysisDecember 1985

Editors:
Roger A. Horn
The Johns Hopkins Univ., Baltimore, MD
,
Charles R. Johnson
Clemson Univ., Clemson, SC

Publisher:

Cambridge University Press
40 W. 20 St. New York, NY
United States

ISBN:978-0-521-30586-0

Published:01 December 1985

Pages:

561

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Roger A Horn
- Publication Years1986 - 2012
- Publication counts8
- Citation count938
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Charles R Johnson
William & Mary
- Publication Years1983 - 2022
- Publication counts45
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Matrix analysis

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Reviews

Reviewer: Andy Roy Magid

This book is a reference and text covering those aspects of matrix theory which are necessary for advanced work in numerical analysis, numerical linear algebra, and applied mathematics. This is by no means a book for beginners: Besides the technical prerequisites of elementary analysis, the authors assume the reader is used to the style of upper-division and graduate-applied mathematics courses. This is also not a book about numerical linear algebra; linear equations hardly occur, and algorithms and their coding get scant mention. But for what the authors intend to do, the book is quite effective. The topics covered—eigenproblems, unitary equivalence, canonical forms, Hermitian matrices, norms, location and perturbation of eigenvalues, positive definite matrices, and nonnegative matrices—are developed in a concise modern style and with complete proofs. Specifically, the book is organized as follows: The first four chapters are the basic core of a linear algebra course, while the remaining five chapters are essentially independent treatments of selected topics. The first chapter, Chapter 0, reviews elementary linear algebra—vector spaces, matrices, determinants. Chapter 1 covers eigenvectors, eigenvalues, and similarity. Chapter 2, on unitary equivalence and normal matrices, deals with unitary triangularization and diagonalization. Chapter 3 covers canonical forms including the Jordan form and its applications. Chapter 4, the first of the independent topics, deals with Hermitian and complex symmetric matrices, with emphasis on variational methods for eigenvalues. Chapter 5 covers norms on vectors and matrices and their properties. Chapter 6 deals with eigenvalue location and perturbation results, including the theory of Gersgorin regions. Chapter 7 is on positive definite matrices and their applications, including inequalities and polar and singular value decompositions. Chapter 8 is about component-wise non-negative and positive matrices and their applications. This is exactly the material that instructors of graduate-applied mathematics courses would wish their students to know. The wealth of material means that instructors using the book as a text will have to skip much of the contents in any given course. But there seems little doubt that the book will become a standard reference for research workers in numerical mathematics.

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Fast Structured Matrix Computations: Tensor Rank and Cohn---Umans Method

"Wide or tall" and "sparse matrix dense matrix" multiplications

Generalized matrix inversion is not harder than matrix multiplication

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Recommendations

Fast Structured Matrix Computations: Tensor Rank and Cohn---Umans Method

"Wide or tall" and "sparse matrix dense matrix" multiplications

Generalized matrix inversion is not harder than matrix multiplication

Reviews

Access critical reviews of Computing literature here