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Perturbation and Intervals of Totally Nonnegative Matrices and Related Properties of Sign Regular Matrices

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Perturbation and Intervals of Totally Nonnegative Matrices and Related Properties of Sign Regular Matrices Gedruckt mit Unterstützung des Deutschen Akademischen Austauschdienstes Perturbation and Intervals of Totally Nonnegative Matrices and Related Properties of Sign Regular Matrices Dissertation zur Erlangung des akademischen Grades des Doktors der Naturwissenschaften (Dr. rer. nat.) an der Mathematisch-Naturwissenschaftliche Sektion Fachbereich Mathematik und Statistik vorgelegt von Mohammad Adm Tag der mündlichen Prüfung: 22. January 2016 Referent: Prof. Dr. Jürgen Garloff, University of Konstanz Referent: Prof. Dr. Shaun Fallat, University of Regina Konstanzer Online-Publikations-System (KOPS) URL: http://nbn-resolving.de/urn:nbn:de:bsz:352-0-334132 Abstract A real matrix is called sign regular if all of its minors of the same order have the same sign or are allowed to vanish and is said to be totally nonnegative and totally positive if all of its minors are nonnegative and positive, respectively. Such matrices arise in a remarkable variety of ways in mathematics and many areas of its applications such as differential and integral equations, function theory, approximation theory, matrix analysis, combinatorics, reductive Lie groups, numerical mathematics, statistics, computer aided geometric design, mathematical finance, and mechanics. A great deal of papers is devoted to the problem of determining whether a given n-by-n matrix is totally nonnegative or totally positive. One could calculate each of its minors, but √ that would involve evaluating about 4n/ πn subdeterminants for a given n-by-n matrix. The question arises whether there is a smaller collection of minors whose nonnegativity or positivity implies the nonnegativity or positivity, respectively, of all minors. In this thesis we derive a condensed form of the Cauchon Algorithm which provides an efficient criterion for total nonnegativity of a given matrix and give an optimal determinantal test for total nonnegativity. By this test we reduce the number of needed minors to only n2. It has long been conjectured, by the supervisor of the thesis, that each matrix in a matrix interval with respect to the checkerboard partial ordering is nonsingular totally nonnegative if the two corner matrices are so, i.e., the so-called interval property holds. Invariance of the totally nonnegativity under element-wise perturbation lays at the core of this conjec- ture. The conjecture was affirmatively answered for the totally positive matrices and some subclasses of the totally nonnegative matrices. In this thesis we settle the conjecture and solve the perturbation problem for totally nonnegative matrices under perturbation of their single entries. The key in settling the conjecture is that the entries of the matrix that is obtained by the application of the Cauchon Algorithm to a given nonsingular totally non- negative matrix can be represented as ratios of two minors formed from consecutive rows and columns of the given matrix. Several analogous results hold for nonsingular totally nonpositive matrices, i.e., matrices having all of their minors nonpositive. By using the Cauchon Algorithm we derive a char- acterization of these matrices and an optimal determinantal test for their recognition. We also prove that these matrices and the nonsingular almost strictly sign regular matrices, a subclass of the sign regular matrices, possess the interval property. These and related results evoke the (open) question whether the interval property holds for general nonsingular sign regular matrices. A partial totally nonnegative (positive) matrix has specified and unspecified entries, and all minors consisting of specified entries are nonnegative (positive). A totally nonnegative (positive) completion problem asks which partial totally nonnegative (positive) matrices al- low a choice of the unspecified entries such that the resulting matrix is totally nonnegative i (positive). Much is known about the totally nonnegative completion problem which is eas- ier to solve than the totally positive completion problem, while very little is known about the latter problem. In this thesis we define new patterns of partial totally positive matrices which are totally positive completable. These patterns settle partially two conjectures which are recently posed by Johnson and Wei. Finally, a real polynomial is called Hurwitz or stable if all of its roots have negative real parts. The importance of the Hurwitz polynomials arises in many scientific fields, for example in control theory and dynamical system theory. A rational function is called an R-function of negative type if it maps the open upper complex half plane into the open lower half plane. With these kinds of polynomials and rational functions some special structured matrices are associated which enjoy important properties like total nonnegativity. Con- versely, from the total nonnegativity of these matrices several properties of polynomials and rational functions associated with these matrices can be inferred. We use these connections to give new and simple proofs of some known facts, prove some properties of these struc- tured matrices, and derive new results for interval polynomials and ”interval” R-functions of negative type. ii Zusammenfassung Eine reelle Matrix wird zeichenfest genannt, wenn alle ihre Minoren gleicher Ordnung das- selbe Vorzeichen besitzen oder verschwinden; sie heißt total nichtnegativ bzw. total positiv, wenn ihre s¨amtlichen Minoren nichtnegativ bzw. positiv sind. Derartige Matrizen treten in einer bemerkenswerten Vielzahl von Gebieten der Mathematik und ihren Anwendungen auf, so z. B. in den Differential- und Integralgleichungen, Funktionen-, Approximations- und Matrizentheorie, Kombinatorik, reduktiven Lie-Gruppen, Numerischer Mathematik, Statis- tik, Computer Aided Geometric Design, Mathematischen Finanz¨okonomie und Mechanik. Eine große Anzahl von Arbeiten besch¨aftigt sich mit dem Problem zu entscheiden, ob eine gegebene quadratische Matrix n-ter Ordnung total nichtnegativ (positiv) ist. Man k¨onnte jeden Minor der Matrix berechnen, aber das wurde¨ die Auswertung von ungef¨ahr √ 4n/ πn Unterdeterminanten der gegebenen Matrix erfordern. Die Frage erhebt sich, ob es eine geringere Anzahl von Minoren gibt, von deren Nichtnegativit¨at bzw. Positivit¨at auf die Nichtnegativit¨at bzw. Positivit¨at s¨amtlicher Minoren der Matrix geschlossen werden kann. In der vorliegenden Arbeit geben wir eine komprimierte Form des Cauchon-Algorithmus an, die ein effektives Kriterium fur¨ die totale Nichtnegativit¨at einer Matrix erm¨oglicht, und leiten einen optimalen Determinantentest fur¨ den Nachweis der totalen Nichtnegativit¨at her. Mit Hilfe dieses Testes l¨asst sich die Anzahl der zu untersuchenden Minoren auf nur n2 reduzieren. Lange Zeit wurde von dem Betreuer dieser Arbeit vermutet, dass jede Matrix aus einem Matrixintervall bezuglich¨ der Schachbrett-Halbordnung regul¨ar und total nichtnegativ ist, wenn die beiden Eckmatrizen des Intervalls diese Eigenschaft besitzen, d. h. die sogenannte Intervall-Eigenschaft gilt. Die Frage der Erhaltung der totalen Nichtnegativit¨at einer Matrix bei St¨orung ihrer einzelnen Koeffizienten ist hiermit eng verknupft.¨ Die Vermutung wurde best¨atigt fur¨ die total positiven Matrizen und einige Unterklassen der total nichtnegativen Matrizen. In der vorliegenden Arbeit beweisen wir die Vermutung und l¨osen das Problem der Erhaltung der totalen Nichtnegativit¨at bei St¨orung einzelner Matrixkoeffizienten. Der Schlussel¨ zum Beweis der Vermutung ist der Sachverhalt, dass die Koeffizienten der Matrix, die mittels Anwendung des Cauchon-Algorithmus auf eine regul¨are und totalnichtnegative Matrix gewonnen wird, dargestellt werden k¨onnen als Quotient zweier Minoren, die aus aufeinanderfolgenden Zeilen und Spalten der Ausgangsmatrix gebildet werden. Verschiedene entsprechende Ergebnisse sind fur¨ die regul¨aren total nichtpositiven Ma- trizen gultig,¨ d. h. fur¨ Matrizen, deren s¨amtliche Minoren nichtpositiv sind. Mit Hilfe des Cauchon-Algorithmus leiten wir eine Charakterisierung dieser Matrizen und einen op- timalen Determinantentest zu ihrer Erkennung her. Wir zeigen ferner, dass diese Matrizen und die regul¨aren fast streng zeichenfesten1) (almost strictly sign regular) Matrizen, eine 1) Wenn uns ein entsprechender deutscher Begriff aus der Literatur nicht bekannt ist, verwenden wir die Ubersetzung¨ des englischen Begriffs und geben diesen in Klammen an. iii Unterklasse der zeichenfesten Matrizen, die Intervall-Eigenschaft besitzen. Diese und ver- wandte Resultate legen die (offene) Frage nahe, ob die Intervall-Eigenschaft auch allgemein fur¨ die regul¨aren zeichenfesten Matrizen gilt. Eine total nichtnegative bzw. positive Teilmatrix (partial matrix) ist eine Matrix, die sowohl spezifizierte als auch unspezifizierte Koeffizienten besitzt, wobei die Minoren, die nur aus spezifizierten Koeffizienten gebildet werden, s¨amtlich nichtnegativ bzw. positiv sind. Ein total nichtnegatives bzw. positives Vervollstandigungsproblem¨ (completion problem) fragt, welche total nichtnegativen bzw. positiven Teilmatrizen eine Wahl s¨amtlicher ihrer un- spezifizierten Koeffizienten erlauben derart, dass die jeweilige vervollst¨andigte Matrix total nichtnegativ bzw. positiv ist. Uber¨ das total nichtnegative Vervollst¨andigungsproblem sind viele Ergebnisse bekannt; dieses ist einfacher zu l¨osen als das total positive Problem, uber¨ welches erst sehr wenige Resultate vorliegen. In dieser Arbeit geben wir neue Muster von total positiven Teilmatrizen an, welche eine total positive Vervollst¨andigung erlauben. Diese Muster best¨atigen teilweise zwei Vermutungen
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