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Partially Specified Matrices and Operators: Classification, Completion, Applications

Author: Israel Gohberg

Publisher: Birkhäuser

Published: 1995-08-28

Total Pages: 368

ISBN-13: 9783764352592

This book is devoted to a new direction in linear algebra and operator theory that deals with the invariants of partially specified matrices and operators, and with the spectral analysis of their completions. The theory developed centers around two major problems concerning matrices of which part of the entries are given and the others are unspecified. The first is a classification problem and aims at a simplification of the given part with the help of admissible similarities. The results here may be seen as a far reaching generalization of the Jordan canonical form. The second problem is called the eigenvalue completion problem and asks to describe all possible eigenvalues and their multiplicities of the matrices which one obtains by filling in the unspecified entries. Both problems are also considered in an infinite dimensional operator framework. A large part of the book deals with applications to matrix theory and analysis, namely to stabilization problems in mathematical system theory, to problems of Wiener-Hopf factorization and interpolation for matrix polynomials and rational matrix functions, to the Kronecker structure theory of linear pencils, and to non everywhere defined operators. The eigenvalue completion problem has a natural associated inverse, which appears as a restriction problem. The analysis of these two problems is often simpler when a solution of the corresponding classification problem is available.

Partially Specified Matrices and Operators: Classification, Completion, Applications

Author: Israel Gohberg

Publisher: Birkhäuser

Published: 2012-12-06

Total Pages: 337

ISBN-13: 3034891008

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Separable Type Representations of Matrices and Fast Algorithms

Author: Yuli Eidelman

Publisher: Springer Science & Business Media

Published: 2013-10-08

Total Pages: 404

ISBN-13: 303480606X

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This two-volume work presents a systematic theoretical and computational study of several types of generalizations of separable matrices. The main attention is paid to fast algorithms (many of linear complexity) for matrices in semiseparable, quasiseparable, band and companion form. The work is focused on algorithms of multiplication, inversion and description of eigenstructure and includes a large number of illustrative examples throughout the different chapters. The first volume consists of four parts. The first part is of a mainly theoretical character introducing and studying the quasiseparable and semiseparable representations of matrices and minimal rank completion problems. Three further completions are treated in the second part. The first applications of the quasiseparable and semiseparable structure are included in the third part where the interplay between the quasiseparable structure and discrete time varying linear systems with boundary conditions play an essential role. The fourth part contains factorization and inversion fast algorithms for matrices via quasiseparable and semiseparable structure. The work is based mostly on results obtained by the authors and their coauthors. Due to its many significant applications and the accessible style the text will be useful to engineers, scientists, numerical analysts, computer scientists and mathematicians alike.

Structured Matrices in Mathematics, Computer Science, and Engineering II

Author: Vadim Olshevsky

Publisher: American Mathematical Soc.

Published: 2001

Total Pages: 362

ISBN-13: 0821820923

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"The collection of the contributions to these volumes offers a flavor of the plethora of different approaches to attack structured matrix problems. The reader will find that the theory of structured matrices is positioned to bridge diverse applications in the sciences and engineering, deep mathematical theories, as well as computational and numberical issues. The presentation fully illustrates the fact that the technicques of engineers, mathematicisn, and numerical analysts nicely complement each other, and they all contribute to one unified theory of structured matrices"--Back cover.

Israel Gohberg and Friends

Author: Harm Bart

Publisher: Springer Science & Business Media

Published: 2008-09-25

Total Pages: 312

ISBN-13: 3764387343

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Mathematicians do not work in isolation. They stand in a long and time honored tradition. They write papers and (sometimes) books, they read the publications of fellow workers in the ?eld, and they meet other mathematicians at conferences all over the world. In this way, in contact with colleagues far away and nearby, from the past (via their writings) and from the present, scienti?c results are obtained whicharerecognizedasvalid.Andthat–remarkablyenough–regardlessofethnic background, political inclination or religion. In this process, some distinguished individuals play a special and striking role. They assume a position of leadership. They guide the people working with them through uncharted territory, thereby making a lasting imprint on the ?eld. So- thing which can only be accomplished through a combination of rare talents: - usually broad knowledge, unfailing intuition and a certain kind of charisma that binds people together. AllofthisispresentinIsraelGohberg,themantowhomthisbookisdedicated,on theoccasionof his 80thbirthday.This comes to the foregroundunmistakably from the contributions from those who worked with him or whose life was a?ected by him. Gohberg’sexceptionalqualitiesarealsoapparentfromthe articleswritten by himself, sometimes jointly with others, that are reproduced in this book. Among these are stories of his life, some dealing with mathematical aspects, others of a more general nature. Also included are reminiscences paying tribute to a close colleaguewho isnotamongusanymore,speechesorreviewshighlightingthework and personality of a friend or esteemed colleague, and responses to the laudatio’s connected with the several honorary degrees that were bestowed upon him.

Factorization of Matrix and Operator Functions: The State Space Method

Author: Harm Bart

Publisher: Springer Science & Business Media

Published: 2007-12-20

Total Pages: 409

ISBN-13: 3764382686

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This book delineates the various types of factorization problems for matrix and operator functions. The problems originate from, or are motivated by, the theory of non-selfadjoint operators, the theory of matrix polynomials, mathematical systems and control theory, the theory of Riccati equations, inversion of convolution operators, and the theory of job scheduling in operations research. The book presents a geometric principle of factorization which has its origins in the state space theory of linear input-output systems and in the theory of characteristic operator functions.

Invariant Subspaces of Matrices with Applications

Author: Israel Gohberg

Publisher: SIAM

Published: 2006-03-01

Total Pages: 706

ISBN-13: 089871608X

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This unique book addresses advanced linear algebra using invariant subspaces as the central notion and main tool. It comprehensively covers geometrical, algebraic, topological, and analytic properties of invariant subspaces, laying clear mathematical foundations for linear systems theory with a thorough treatment of analytic perturbation theory for matrix functions.

Recent Developments in Operator Theory and Its Applications

Author: I. Gohberg

Publisher: Birkhäuser

Published: 2012-12-06

Total Pages: 448

ISBN-13: 3034890354

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The papers selected for publication here, many of them written by leaders in the field, bring readers up to date on recent achievements in modern operator theory and applications. The book’s subject matter is of practical use to a wide audience in mathematical and engineering sciences.

The Extended Field of Operator Theory

Author: Michael A. Dritschel

Publisher: Springer Science & Business Media

Published: 2007-06-25

Total Pages: 405

ISBN-13: 3764379804

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This volume contains contributions originating from the International Workshop on Operator Theory and Its Applications (IWOTA) held in Newcastle upon Tyne in July 2004. The articles expertly cover a broad range of material at the cutting edge of functional analysis and its applications. The works are written by world authorities in their specialities.

Operator Theory, Analysis and the State Space Approach

Author: Harm Bart

Publisher: Springer

Published: 2018-12-30

Total Pages: 464

ISBN-13: 3030042693

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This volume is dedicated to Rien Kaashoek on the occasion of his 80th birthday and celebrates his many contributions to the field of operator theory during more than fifty years. In the first part of the volume, biographical information and personal accounts on the life of Rien Kaashoek are presented. Eighteen research papers by friends and colleagues of Rien Kaashoek are included in the second part. Contributions by J. Agler, Z.A. Lykova, N.J. Young, J.A. Ball, G.J. Groenewald, S. ter Horst, H. Bart, T. Ehrhardt, B. Silbermann, J.M. Bogoya, S.M. Grudsky, I.S. Malysheva, A. Böttcher, E. Wegert, Z. Zhou, Y. Eidelman, I. Haimovici, A.E. Frazho, A.C.M. Ran, B. Fritzsche, B. Kirstein, C.Madler, J. J. Jaftha, D.B. Janse van Rensburg, P. Junghanns, R. Kaiser, J. Nemcova, M. Petreczky, J.H. van Schuppen, L. Plevnik, P. Semrl, A. Sakhnovich, F.-O. Speck, S. Sremac, H.J. Woerdeman, H. Wolkowicz and N. Vasilevski.