Matrix Mathematics: Theory, Facts, and Formulas - Softcover

Bernstein, Dennis S.

Softcover

ISBN 10: 0691140391 ISBN 13: 9780691140391

Verlag: Princeton University Press, 2009

Inhaltsangabe

When first published in 2005, Matrix Mathematics quickly became the essential reference book for users of matrices in all branches of engineering, science, and applied mathematics. In this fully updated and expanded edition, the author brings together the latest results on matrix theory to make this the most complete, current, and easy-to-use book on matrices.


Each chapter describes relevant background theory followed by specialized results. Hundreds of identities, inequalities, and matrix facts are stated clearly and rigorously with cross references, citations to the literature, and illuminating remarks. Beginning with preliminaries on sets, functions, and relations,Matrix Mathematics covers all of the major topics in matrix theory, including matrix transformations; polynomial matrices; matrix decompositions; generalized inverses; Kronecker and Schur algebra; positive-semidefinite matrices; vector and matrix norms; the matrix exponential and stability theory; and linear systems and control theory. Also included are a detailed list of symbols, a summary of notation and conventions, an extensive bibliography and author index with page references, and an exhaustive subject index. This significantly expanded edition of Matrix Mathematics features a wealth of new material on graphs, scalar identities and inequalities, alternative partial orderings, matrix pencils, finite groups, zeros of multivariable transfer functions, roots of polynomials, convex functions, and matrix norms.

• Covers hundreds of important and useful results on matrix theory, many never before available in any book


• Provides a list of symbols and a summary of conventions for easy use


• Includes an extensive collection of scalar identities and inequalities


• Features a detailed bibliography and author index with page references


• Includes an exhaustive subject index with cross-referencing

Über diesen Titel

Über die Autorin bzw. den Autor

Dennis S. Bernstein is professor of aerospace engineering at the University of Michigan.

Auszug. © Genehmigter Nachdruck. Alle Rechte vorbehalten.

Matrix Mathematics

Theory, Facts, and FormulasBy Dennis S. Bernstein

PRINCETON UNIVERSITY PRESS

Copyright © 2009 Princeton University Press
All right reserved.
ISBN: 978-0-691-14039-1

Contents

Preface to the Second Edition..............................................xvPreface to the First Edition...............................................xviiSpecial Symbols............................................................xxiConventions, Notation, and Terminology.....................................xxxiii1. Preliminaries...........................................................12. Basic Matrix Properties.................................................853. Matrix Classes and Transformations......................................1794. Polynomial Matrices and Rational Transfer Functions.....................2535. Matrix Decompositions...................................................3096. Generalized Inverses....................................................3977. Kronecker and Schur Algebra.............................................4398. Positive-Semidefinite Matrices..........................................4599. Norms...................................................................59710. Functions of Matrices and Their Derivatives............................68111. The Matrix Exponential and Stability Theory............................707Bibliography...............................................................881Author Index...............................................................967Index......................................................................979

Chapter One

Preliminaries

In this chapter we review some basic terminology and results concerning logic, sets, functions, and related concepts. This material is used throughout the book.

1.1 Logic

Every statement is either true or false, but not both. Let A and B be statements. The negation of A is the statement (not A), the both of A and B is the statement (A and B), and the either of A and B is the statement (A or B). The statement (A or B) does not contradict (A and B), that is, the word "or" is inclusive. Exclusive "or" is indicated by the phrase "but not both."

The statements "A and B or ITLITL" and "A or B and ITLITL" are ambiguous. We therefore write "A and either B or ITLITL" and "either A or both B and ITLITL."

Let A and B be statements. The implication statement "if A is satisfied, then B is satisfied" or, equivalently, "A implies B" is written as A [??] B, while A [??] B is equivalent to [(A [??] B) and (A [??] B)]. Of course, A [??] B means B [??] A. A tautology is a statement that is true regardless of whether the component statements are true or false. For example, the statement "(A and B) implies A" is a tautology. A contradiction is a statement that is false regardless of whether the component statements are true or false. For example, the statement "A implies (not) A" is a contradiction.

Suppose that A [??] B. Then, A is satisfied if and only if B is satisfied. The implication A [??] B (the "only if" part) is necessity, while B [??] A (the "if" part) is sufficiency. The converse statement of A [??] B is B [??] A. The statement A [??] B is equivalent to its contrapositive statement (not B) [??] (not A).

A theorem is a significant statement, while a proposition is a theorem of less significance. The primary role of a lemma is to support the proof of a theorem or proposition. Furthermore, a corollary is a consequence of a theorem or proposition. Finally, a fact is either a theorem, proposition, lemma, or corollary. Theorems, propositions, lemmas, corollaries, and facts are provably true statements.

Suppose that A' [??] A [??] B [??] B'. Then, A' [??] B' is a corollary of A [??] B.

Let A, B, and ITLITL be statements, and assume that A [??] B. Then, A [??] B is a strengthening of the statement (A and ITLITL) [??] B. If, in addition, A [??] C, then the statement (A and ITLITL) [??] B has a redundant assumption.

1.2 Sets

A set {x, y, ...} is a collection of elements. A set may have a finite or infinite number of elements. A finite set has a finite number of elements.

Let X be a set. Then,

x [member of] X (1.2.1)

means that x is an element of X. If w is not an element of X, then we write

w [??] X. (1.2.2)

The statement "x [member of] X" is either true or false, but not both. The statement "X [??] X" is true by convention, and thus no set can be an element of itself. Therefore, there does not exist a set that contains every set. The set with no elements, denoted by Ø, is the empty set. If X ≠ Ø, then X is nonempty.

A set cannot have repeated elements. For example, {x, x} = {x}. However, a multiset is a collection of elements that allows for repetition. The multiset consisting of two copies of x is written as {x, x}_ms. However, we do not assume that the listed elements x, y of the conventional set {x, y} are distinct. The number of distinct elements of the set S or not-necessarily-distinct elements of the multiset S is the cardinality of S, which is denoted by card(S).

There are two basic types of mathematical statements for quantifiers. An existential statement is of the form

there exists x [member of] X such that statement Z is satisfied, (1.2.3)

while a universal statement has the structure

for all x [??] X, it follows that statement Z is satisfied, (1.2.4)

or, equivalently,

statement Z is satisfied for all x [member of] X. (1.2.5)

Let X and Y be sets. The intersection of X and Y is the set of common elements of X and Y given by

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] (1.2.6)

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] (1.2.7)

while the set of elements in either X or Y (the union of X and Y) is

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] (1.2.8)

The complement of X relative to Y is

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] (1.2.9)

If Y is specified, then the complement of X is

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] (1.2.10)

If x [member of] X implies that x [member of] Y, then X is contained in Y (X is a subset of Y), which is written as

X [??] Y. (1.2.11)

The statement X = Y is equivalent to the validity of both X [??] Y and Y [??] X. If X [??] Y and X ≠ Y, then X is a proper subset of Y and we write X [subset] Y. The sets X and Y are disjoint if X [intersection] Y = Ø. A partition of X is a set of pairwise-disjoint and nonempty subsets of X whose union is equal to...

Details zur Ausgabe

• Verlag: Princeton University Press
• Erscheinungsdatum: 2009
• Sprache: Englisch
• ISBN 10: 0691140391
• ISBN 13: 9780691140391
• Einband: Taschenbuch
• Auflage: 2
• Anzahl der Seiten: 1139
• Kontakt zum Hersteller: Nicht verfügbar
• Verantwortliche Person: Nicht verfügbar