Matrix Theory: From Generalized Inverses to Jordan Form (Chapman & Hall CRC Pure and Applied Mathematics)

569 Pages·2007·6.22 MB·English

by  Robert Piziak

Matrix Theory
From Generalized Inverses
to Jordan Form

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Matrix Theory
From Generalized Inverses
to Jordan Form
Robert Piziak
Baylor University
Texas, U.S.A.
P. L. NO
Baylor University
Texas, U.S.A.
Chapman & Hall/CRC
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Library of Congress Cataloging-in-Publication Data
Piziak, Robert.
Matrix theory : from generalized inverses to Jordan form / Robert Piziak and P.L.
Odell
p cm -- (Pure and applied mathematics)
Includes bibliographical references and index.
ISBN-13.978-1-5848 8-625-9 (acid-free paper)
1. Matrices--textbooks. 2. Algebras, Linerar--Textbooks. 3. Matrix
inversion--Textbooks. I.Odell, Patrick L., 1930- II Title. Ill. Series.
QA 188. P59 2006
512.9'434--dc22 2006025707
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Preface
This text is designed for a second course in matrix theory and linear algebra
accessible to advanced undergraduates and beginning graduate students. Many
concepts from an introductory linear algebra class are revisited and pursued to
a deeper level. Also, material designed to prepare the student to read more ad-
vanced treatises and journals in this area is developed. A key feature of the book
is the idea of "generalized inverse" of a matrix, especially the Moore-Penrose
inverse. The concept of "full rank factorization" is used repeatedly throughout
the book. The approach is always "constructive" in the mathematician's sense.
The important ideas needed to prepare the reader to tackle the literature in
matrix theory included in this book are the Henderson and Searle formulas,
Schur complements, the Sherman-Morrison-Woodbury formula, the LU fac-
torization, the adjugate, the characteristic and minimal polynomial, the Frame
algorithm and the Cayley-Hamilton theorem, Sylvester's rank formula, the fun-
damental subspaces of a matrix, direct sums and idempotents, index and the
Core-Nilpotent factorization, nilpotent matrices, Hermite echelon form, full
rank factorization, the Moore-Penrose inverse and other generalized inverses,
norms, inner products and the QR factorization, orthogonal projections, the
spectral theorem, Schur's triangularization theorem, the singular value decom-
position, Jordan canonical form, Smith normal form, and tensor products.
This material has been class tested and has been successful with students
of mathematics, undergraduate and graduate, as well as graduate students in
statistics and physics. It can serve well as a "bridge course" to a more advanced
study of abstract algebra and to reading more advanced texts in matrix theory.
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