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

(1)

(2)

Gedruckt mit Unterstützung des Deutschen Akademischen Austauschdienstes

(3)

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

(4)

(5)

Abstract

A real matrix is calledsign 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 nonnegativeand 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 givenn-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 onlyn².

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 conjecture. 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 nonnegative 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 characterization 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 problemasks which partial totally nonnegative (positive) matrices al- low a choice of the unspecified entries such that the resulting matrix is totally nonnegative

(6)

(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 typeif 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 structured matrices, and derive new results for interval polynomials and ”interval” R-functions of negative type.

(7)

Zusammenfassung

Eine reelle Matrix wirdzeichenfestgenannt, wenn alle ihre Minoren gleicher Ordnung das- selbe Vorzeichen besitzen oder verschwinden; sie heißt total nichtnegativbzw. 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 beschäftigt sich mit dem Problem zu entscheiden, ob eine gegebene quadratische Matrix n-ter Ordnung total nichtnegativ (positiv) ist. Man könnte jeden Minor der Matrix berechnen, aber das würde die Auswertung von ungefähr 4ⁿ/√

πnUnterdeterminanten der gegebenen Matrix erfordern. Die Frage erhebt sich, ob es eine geringere Anzahl von Minoren gibt, von deren Nichtnegativität bzw. Positivität auf die Nichtnegativität bzw. Positivität sämtlicher Minoren der Matrix geschlossen werden kann.

In der vorliegenden Arbeit geben wir eine komprimierte Form des Cauchon-Algorithmus an, die ein effektives Kriterium für die totale Nichtnegativität einer Matrix ermöglicht, und leiten einen optimalen Determinantentest für den Nachweis der totalen Nichtnegativität her. Mit Hilfe dieses Testes lässt 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 bezüglich der Schachbrett-Halbordnung regulär und total nichtnegativ ist, wenn die beiden Eckmatrizen des Intervalls diese Eigenschaft besitzen, d. h. die sogenannte Intervall-Eigenschaftgilt. Die Frage der Erhaltung der totalen Nichtnegativität einer Matrix bei Störung ihrer einzelnen Koeffizienten ist hiermit eng verknüpft. Die Vermutung wurde bestätigt für die total positiven Matrizen und einige Unterklassen der total nichtnegativen Matrizen. In der vorliegenden Arbeit beweisen wir die Vermutung und lösen das Problem der Erhaltung der totalen Nichtnegativität bei Störung einzelner Matrixkoeffizienten. Der Schlüssel zum Beweis der Vermutung ist der Sachverhalt, dass die Koeffizienten der Matrix, die mittels Anwendung des Cauchon-Algorithmus auf eine reguläre und totalnichtnegative Matrix gewonnen wird, dargestellt werden können als Quotient zweier Minoren, die aus aufeinanderfolgenden Zeilen und Spalten der Ausgangsmatrix gebildet werden.

Verschiedene entsprechende Ergebnisse sind für die regulären total nichtpositiven Ma- trizen gültig, d. h. für Matrizen, deren sämtliche Minoren nichtpositiv sind. Mit Hilfe des Cauchon-Algorithmus leiten wir eine Charakterisierung dieser Matrizen und einen optimalen Determinantentest zu ihrer Erkennung her. Wir zeigen ferner, dass diese Matrizen und die regulären 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.¨

(8)

Unterklasse der zeichenfesten Matrizen, die Intervall-Eigenschaft besitzen. Diese und ver- wandte Resultate legen die (offene) Frage nahe, ob die Intervall-Eigenschaft auch allgemein f¨ur 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, sämtlich nichtnegativ bzw. positiv sind. Ein total nichtnegatives bzw. positives Vervollständigungsproblem (completion problem) fragt, welche total nichtnegativen bzw. positiven Teilmatrizen eine Wahl sämtlicher ihrer un- spezifizierten Koeffizienten erlauben derart, dass die jeweilige vervollständigte Matrix total nichtnegativ bzw. positiv ist. Über das total nichtnegative Vervollständigungsproblem sind viele Ergebnisse bekannt; dieses ist einfacher zu lösen als das total positive Problem, über welches erst sehr wenige Resultate vorliegen. In dieser Arbeit geben wir neue Muster von total positiven Teilmatrizen an, welche eine total positive Vervollständigung erlauben. Diese Muster bestätigen teilweise zwei Vermutungen aus einer kürzlich erschienenen Arbeit von Johnson und Wei.

Ein Polynom heißt (Hurwitz-)stabil, wenn sämtliche seiner Nullstellen negativen Realteil besitzen. Derartige Polynome treten in vielen Gebieten auf, so z. B. in der Regelungstheorie und bei dynamischen Systemen. Eine rationale Funktion wird eine R-Funktion von neg- ativem Typ genannt, wenn sie die obere offene komplexe Halbebene auf die untere offene Halbebene abbildet. Mit den stabilen Polynomen und den R-Funktionen vom negativen Typ sind strukturierte Matrizen verknüpft, die wichtige Eigenschaften wie die der totalen Nichtnegativität besitzen. Umgekehrt können aus der totalen Nichtnegativität dieser Ma- trizen verschiedene Eigenschaften der zugrundeliegenden Polynome und rationalen Funk- tionen gefolgert werden. Wir verwenden diese Beziehungen, um neue und einfache Beweise für einige bekannte Sachverhalte anzugeben, Eigenschaften dieser strukturierten Matrizen zu beweisen und neue Ergebnisse für Intervallpolynome und ”Intervall”-R-Funktionen vom negativen Typ herzuleiten.

(9)

Acknowledgements

I would like to thank all the people who had helped and supported me during all the stages of this work.

First and foremost, I would like to express my deep appreciation and warm thanks to my supervisor Prof. Dr. Jürgen Garloff for his expert, friendly, inspiring guidance and co- operation, as well as his patience and generosity. I would like to thank him for continuous encouragement and supporting me to grow as a research scientist. I would like also to thank Prof. Dr. Shaun M. Fallat for serving as the second reviewer and for his valuable comments and suggestions and Prof. Dr. Salma Kuhlmann and Prof. Dr. Michael Junk for serving as committee members.

I am very grateful to the funding source that made my PhD work possible, the German Academic Exchange Service (DAAD). For their help and advices, I would like to thank Prof.

Dr. Helga Baumgarten, Mrs. Saoussen Louti, and Mrs. Eveline Muhareb.

My special thanks go to Prof. Dr. Georg Umlauf (HTWG/Konstanz) for his valuable help, especially for writing reviews for the DAAD, and to the HTWG/Konstanz for its hos- pitality. Many thanks go to Prof. Dr. Fuad Kittaneh (The University of Jordan) and Dr.

Andrew P. Smith.

I also would like to express my sincere gratitude to the Welcome Center at the University of Konstanz and Mr. Rainer Janßen from the Mathematics and Statistics Department for their valuable help in many regards.

I am deeply indebted to Palestine Polytechnic University, in Hebron, Palestine, especially to Ms. Khawla Al-Muhtaseb, Dr. Ali Zein, Mr. Ayed Abed El-Gani Head of the Applied Mathematics and Physics Department, Prof. Dr. Ibrahim Masri, Dr. Mustafa Abu Safa, Vice President for Academic Affairs, and Dr. Amjad Barham, Vice President for Adminis- trative Affairs, for their continuous interest and encouragement.

A special thank to my family. Words cannot express how grateful I am to my father, mother, wife, brothers, sisters and their husbands, mother-in-law, and father-in-law for all of the sacrifices that you have made on my behalf. Your prayer for me was what sustained me thus far. I would like also to thank all of my friends who encouraged me to strive towards my goal in Konstanz and abroad especially Mr. Nael Abukhiran. Special thanks and appreciations go to Mrs. Chistiane Hauber-Garloff for her kindness and generosity. At the end I would like to express my gratitude to my beloved wife who gives me love and tenderness. She has been one of my supports at all of the moments during the last three years. Special thanks to her for designing the cover page.

(10)

List of Figures

2.1. Elementary diagrams for D (left), L_k(l) (middle), and U_j(u) (right), [FJ11,

Figure 2.1] . . . 30

2.2. Planar network Γ₀ . . . 31

3.1. An example of a Cauchon diagram . . . 41

3.2. Condition 3.(a) of Definition 3.4 . . . 47

3.3. Condition 3.(b) of Definition 3.4 . . . 48

(13)

1. Introduction

Several types of matrices play an important role in various branches of mathematics and other sciences. A particular instance of these matrices are the sign regular matrices which appear in remarkably diverse areas of mathematics and its applications, including approximation theory, numerical mathematics, statistics, economics, computer aided geometric design, and other fields, cf. [GK60], [Kar68], [And87], [GM96], [P99], [Pin10], [FJ11], [HT12].

A real matrix is calledsign regularandstrictly sign regularif all its minors of the same order have the same sign or vanish and are nonzero and have the same sign, respectively. If the sign of all minors of any order is nonnegative (positive, nonpositive, and negative) then the matrix is calledtotally nonnegative(totally positive, totally nonpositive, and totally negative, respectively). In the existing literature, the terminologies of totally nonnegative and totally positive are not consistent with that we are using throughout the thesis. Elsewhere in the literature the terms totally nonnegative and totally positive correspond to totally positive and strictly totally positive, respectively, see, e.g., [Kar68], [And87], [GM96], [Pin10].

In this thesis we consider the following problems:

• The condensed form of the Cauchon Algorithm, characterizations of totally nonnegative matrices and nonsingular totally nonnpositive matrices by using the Cauchon Algorithm, relationship between Neville elimination and the Cauchon Algorithm, representations of the entries of the matrix that is obtained by the application of the Cauchon Algorithm to a nonsingular totally nonnegative or a nonsingular totally nonpositive matrix, optimal determinantal criteria for total nonnegativity and nonsingular total nonpositivity, and characterization of several subclasses of totally nonnegative matrices by using the Cauchon Algorithm.

• Matrix intervals of nonsingular sign regular matrices with respect to the checkerboard ordering.

• Invariance of total nonnegativity under element-wise perturbation and total nonnegativity of the extended Perron complement.

• Totally nonnegative and totally positive completion problems.

• Total nonnegativity of special structured matrices, stability of given polynomials and location of zeros and poles of rational functions by using total nonnegativity of certain matrices, stability of interval polynomials, and interval problems for a subclass of the rational functions, viz. theR-functions.

(14)

1.1. Overview

This thesis is divided into eight chapters. In the following we give a brief overview of each chapter.

Chapter 1: Most of the definitions, notations, and determinantal identities and inequalities that will be used throughout the thesis are introduced.

Chapter 2: We present several criteria and properties of sign regular, strictly sign regular, almost strictly sign regular, totally nonpositive, totally negative, totally positive, and totally nonnegative matrices and some of their subclasses. The variation diminishing property, Neville elimination, and planar networks and their relations to totally nonnegative matrices are described. We also give some spectral properties of some subclasses of the sign regular matrices.

Chapter 3: We introduce the totally nonnegative cells and the Cauchon Algorithm.

We study the totally nonnegative and totally nonnpositive matrices through the Cauchon Algorithm. Hereby determinantal criteria for totally nonnegative matrices and nonsingular totally nonpositive matrices, representations of the entries of the matrix that is obtained by the application of the Cauchon Algorithm to a nonsingular totally nonnegative and a nonsingular totally nonpositive matrix, and characterizations of several subclasses of totally nonnegative matrices are derived.

Chapter 4: We present several known results of matrix intervals of sign regular matrices with respect to the checkerboard partial ordering and prove new results, e.g., Garloff’s Conjecture and analogous results for the nonsingular totally nonpositive, nonsingular almost strictly sign regular, nonsingular tridiagonal sign regular, and nonsingular sign regular matrices of special signatures.

Chapter 5: We close the book on the perturbation problem of the single entries of totally nonnegative matrices and extend some known results on the total nonnegativity of the extended Perron complement.

Chapter 6: We review some totally nonnegative and totally positive completion problems and present several new results which lead us to settle partially two conjectures posed recently.

Chapter 7: We consider the problem of stability of polynomials and introduce several notions and structured matrices which play an important role in the study of polynomials and rational functions.

Chapter 8: We apply our results on totally nonnegative matrices to the matrices that are introduced in Chapter 7 to investigate the relationships between total nonnegativity of

(15)

these matrices and the stability of the associated polynomials and the location of zeros and poles of the associated rational functions. We also apply our results on intervals of totally nonnegative matrices to derive sufficient conditions for the stability of interval polynomials and special ”interval” rational functions, viz. R-functions of negative type. These problems include invariance of exclusively positive poles and exclusively negative roots in the presence of variation of the coefficients of the polynomials within given intervals.

1.2. Definitions and Notation

In this section we introduce most of the definitions and notations that will be used throughout this thesis. All of these definitions are extended verbatim to infinite matrices.

The set of the n-by-m real matrices is denoted by R^n,m. For integers κ, n we denote by Qκ,n the set of all strictly increasing sequences of κ integers chosen from {1,2, . . . , n}.

We use the set theoretic symbols∪and \ to denote somewhat not precisely but intuitively the union and the difference, respectively, of two index sequences, where we consider the resulting sequence as strictly increasing ordered. Forα∈Qκ,nwe defineα^c:={1, . . . , n}\α.

ForA∈R^n,m,α= (α₁, α₂, . . . , α_κ)∈Q_κ,n, andβ = (β₁, β₂, . . . , β_µ)∈Q_µ,m, we denote by A[α|β] theκ-by-µsubmatrix of Alying in the rows indexed by α1, α2, . . . , ακ and columns indexed byβ₁, β₂, . . . , β_µ. We suppress the brackets when we enumerate the indices explic- itly. ByA(α|β) we denote the (n−κ)-by-(m−µ) submatrixA[α^c|β^c] ofA. Whenα=β, the principal submatrixA[α|α] is abbreviated to A[α] and detA[α] is called a principal minor, with the similar notationA(α) for the complementary principal submatrix. In the special case whenα= (1,2, . . . , κ), we refer to the principal submatrixA[α] as theleading principal submatrix(and to detA[α] as theleading principal minor)of order κ. We denote by|α|the number of members of α. A measure of the gaps in an index sequence α is the dispersion of α, denoted by d(α), and is defined to be d(α) := α_κ−α₁−κ+ 1. If d(α) = 0 we call α contiguous, if d(α) =d(β) = 0 we call the submatrix A[α|β] contiguous and in the case κ=µwe call the corresponding minor contiguous.

For α, β ∈Qκ,n, we set αˆi := α\ {i} for some i∈α and detA[α|β] := 1 if α or β is not strictly increasing or empty, in particular, we put detA[α₁, α₂] = 1 if α₁ > α₂ (possibly α₂

= 0).

We order the sequences from Qκ,nwith respect to the lexicographical and colexicographical ordering. We denote by ≤ and ≤_c the lexicographical and colexicographical ordering, respectively, i.e.,

α = (α1, . . . , ακ)≤α^∗ = (α^∗₁, . . . , α^∗_κ)

if and only ifα =α^∗ or the first nonvanishing difference in the following sequence α^∗₁−α1, α^∗₂−α2, . . . , α^∗_κ−ακ

(16)

is positive and

α= (α₁, . . . , α_κ)≤_cα^∗ = (α^∗₁, . . . , α^∗_κ)

if and only ifα =α^∗ or the first nonvanishing difference in the following sequence α^∗_κ−ακ, α^∗_κ−1−ακ−1, . . . , α^∗₁−α1

is positive.

In the remaining chapters we will consider the lexicographical and colexicographical ordering on pairs of indices instead on strictly increasing sequences.

A minor detA[α|β] of A is calledrow-initial ifα= (1,2, . . . , κ) and β∈Qκ,m is contiguous,column-initial ifα∈Q_κ,nis contiguous whileβ = (1,2, . . . , κ),initial if it is row-initial or column-initial, and quasi-initial if either α = (1,2, . . . , κ) and β ∈ Qκ,m is arbitrary or α∈Qκ,n is arbitrary, whileβ = (1,2, . . . , κ).

The identity matrix of ordernis denoted byIn. The n-by-nmatrix whose only nonzero entry is a one in the (i, j)^thposition is denoted byEij. We reserve throughout the notation T_n= (t_ij) for the (anti-diagonal matrix) permutation matrix of ordernwitht_ij =δi,n−j+1, i, j= 1, . . . , n, and set A^#:= TnATm forA ∈R^n,m. A matrix A= (aij)∈R^n,n is referred to as atridiagonal(orJacobi),pentadiagonal,lower triangular, andupper triangularmatrix ifa_ij = 0 whenever |i−j|>1,|i−j|>2,j > i, andi > j, respectively.

An n-by-n matrix A is termed irreducibleif either n= 1 and A 6= 0 orn≥2 and there is no permutation matrixP such that

P AP^T =

"

B C

0 D

# ,

where 0 is the (n−r)-by-r zero matrix (1≤r ≤n−1). Otherwise it is called reducible.

A sequence is termed a signature sequence if all its members are −1 or 1. A matrix A ∈ R^n,m is called strictly sign regular abbreviated as SSR and sign regular SR with signature = (1, . . . , n⁰) if 0 < κdetA[α|β] and 0 ≤ κdetA[α|β], respectively, for all α ∈ Qκ,n, β ∈ Qκ,m, κ = 1,2, . . . , n⁰, where n⁰ := min{n, m}. If A is SSR (SR) with signature= (1,1, . . . ,1), thenA is termed totally positive T P (totally nonnegative T N).

Ais said to betotally positive(totally nonnegative)of order k≤n⁰ T Pk (T Nk) if all its minors of order less than or equal tokare positive (nonnegative). If a square matrixAisT N and has aT P integral power then it is called oscillatory. If Ais SSR (SR) with signature = (−1,−1, . . . ,−1), thenA is called totally negative t.n. (totally nonpositive t.n.p.). If a square matrix A is in a certain class of SR matrices and in addition also nonsingular then we affixN sto the name of the class, i.e., if A is T N and in addition nonsingular we write Ais N sT N. If all the principal minors ofA are positive thenAis referred to as P-matrix.

A real square matrix A is termed positive definite if it is symmetric and a P-matrix. In

(17)

passing we note that ifAisT N then so are its transpose, denoted byA^T, andA^#, see, e.g., [FJ11, Theorem 1.4.1].

A lower (upper) triangular matrix A is called totally positive (abbreviated 4T P) if it is T N and has its minors positive unless they are identical zero because of the triangular structure of A. An n-by-n matrix A is said to have an LU (LDU) factorization if A can be written as A = LU (A = LDU), where L, (D), and U are n-by-n lower triangular, (diagonal), and upper triangular matrices, respectively.

For ann-by-nmatrixAwithA[α] is nonsingular for someα∈Qk,n, theSchur complement of A[α] in A, denoted by A/A[α], is defined as

A/A[α] :=A[α^c]−A[α^c|α](A[α])⁻¹A[α|α^c]. (1.1) Finally, we endowR^n,n with two partial orderings:

Firstly, with the usualentry-wise partial ordering (A= (aij),B = (bij)∈R^n,n) A≤B :⇔ a_ij ≤b_ij, i, j = 1, . . . , n,

and with thecheckerboard partial orderingwhich is defined as follows. LetS:= diag (1,−1, . . . , (−1)ⁿ⁺¹) and A^∗ :=SAS. Then we define

A≤^∗ B :⇔ A^∗ ≤B^∗.

In particular, we say that a real matrixA isnonnegative (positive) if 0≤(<)A.

1.3. Compound Matrices and Kronecker’s Theorem

In this section we present compound matrices, some of their properties, their relation to the sign regular matrices, and Kronecker’s Theorem. For more details, the interested reader is referred to [Kar68], [GK60], [Pin10].

LetA= (aij) be ann-by-m matrix. The sequences inQp,n are considered to be ordered with respect to the lexicographical order. Hence each sequence will occupy a definite place comparable to the other sequences inQp,nand consequently it will have a definite numbers, which can run through the values 1,2, . . . , ⁿ_p. The same process is done for the sequences inQ_p,m.

For the minors of order pof the matrix Aset

a^(p)_st := detA[α|β], (1.2)

wheresand tare the numbers of the sequences αinQ_p,n and β inQ_p,m, respectively. The matrix

A^(p)= (a^(p)_st )^N,M_s=1,t=1, N = n p

!

, M = m

p

!

, p= 1, . . . ,min{n, m},

(18)

is called thep-th compound matrix of A or thep-th associated matrix of A.

Compound matrices play a fundamental role in matrix theory and provide a method of constructing new matrices from given ones.

In order to find all the compound matrices of a givenn-by-mmatrix, one needs to calculate

min{n,m}

X

p=1

n p

! m p

!

= n+m n

!

−1 (1.3)

minors. For n =m this number is of order 4ⁿn^−1/2 (by using Stirling’s approximation to the factorial).

The following theorem gives some properties which are very useful in studying compound matrices.

Theorem 1.1. [Kar68], [Kus12] Let A, B∈R^n,n. Then the following hold:

1. A^(j)= 0 if and only if j ≥r, where r is the rank of A.

2. In^(j)=I(ⁿ_j).

3. (AB)^(j)=A^(j)B^(j).

4. For any positive integer m, (A^(j))^m = (A^m)^(j). 5. If A is nonsingular, then (A^(j))⁻¹= (A⁻¹)^(j).

By using the compound matrix terminology we can restate the definition of SR matrices in terms of the compound matrices. A matrix A ∈ R^n,m is SR (SSR) with signature (1, . . . , _n⁰) if each p-th compound matrix ofA has entry-wise the sign p, where 0 is per- mitted, (strict sign _p), and T N (T P) if all its p-th compound matrices are entry-wise nonnegative (positive),p= 1, . . . , n⁰.

The following theorem due to Kronecker provides the relationship between the eigenvalues of a given matrix and that of itsp-th compound matrices.

Theorem 1.2. [Kar68], [GK02, Theorem 23, p. 65] Let λ₁, λ₂, . . . , λ_n be the complete system of eigenvalues of the matrixA. Then the complete system of eigenvalues of the p-th compound matrix A^(p) consists of all possible products of the numbers λ1, λ2, . . . , λn taken p at a time.

(19)

1.4. Determinantal Identities and Inequalities

In this section we list and briefly describe various determinantal identities which are now indispensable ingredients in matrix theory. Also we give some determinantal inequalities that hold for all totally nonnegative matrices.

We begin with the following fundamental formula in the theory of matrices which is attributed to Cauchy and Binet. This identity can be seen as a generalization of the multiplication formula of matrices and is in the square case identical with property 3. in Theorem 1.1.

Lemma 1.1. [FJ11, Cauchy-Binet Identity, Theorem 1.1.1] LetAbe an m-by-pmatrix and B an p-by-n matrix, and let α ⊆ {1, . . . , m} and β ⊆ {1, . . . , n} be k-element sets with k≤min{n, m, p}. Then

detAB[α|β] =^X

γ

detA[α|γ] detB[γ|β], (1.4)

where γ ranges over Qk,p.

The following determinantal identity due to Sylvester serves as a basis tool in some proofs of this thesis.

Lemma 1.2. [GK02, Theorem 1, p. 13], [Pin10, Sylvester’s Determinant Identity] Let A be an n-by-m matrix such that detA[α|β]6= 0, whereα, β are both in Q_k,n and B = (bij) is the matrix obtained from A by

bij := detA[α∪ {i} |β∪ {j}]

detA[α|β] , for all (i, j)∈α^c×β^c, (1.5) then

detB[η|ζ] = detA[α∪η|β∪ζ]

detA[α|β] , for all η ⊆α^c, ζ⊆β^c. (1.6) The submatrix A[α|β] in Lemma 1.2 is called the pivot block, see, e.g., [Kar68, p. 5], since this identity is proven by observing that B is theSchur complement of A[α|β] in A.

A consequence of Sylvester’s determinant identity is the next corollary.

Corollary 1.1. [dBP82, Corollary 1, p. 84] Let A, B be given as in Lemma 1.2. Then rank (A[α∪α⁰|β∪β⁰]) =|α|+ rank (B[α⁰|β⁰]),

where α∩α⁰ =β∩β⁰=φ.

In the sequel we will often make use of the following special case of Sylvester’s determinant identity, see, e.g., [FJ11, pp. 29-30].

(20)

Lemma 1.3. PartitionA∈R^n,n, n≥3, as follows:

A=







c A₁₂ d

A21 A22 A23

e A32 f





,

where A22∈R^n−2,n−2 andc, d, e, f are scalars. Define the submatrices

C:= c A₁₂

A21 A22

!

, D:= A₁₂ d A22 A23

! ,

E:= A₂₁ A₂₂ e A32

!

, F := A₂₂ A₂₃ A32 f

! . Then ifdetA226= 0, we have

detA= detCdetF −detDdetE

detA₂₂ .

The following lemma relates the minors of a given nonsingular square matrix with the minors of its inverse. In particular, it plays a fundamental role in proving that for a given n-by-n N sT NmatrixA, the matrixSA⁻¹SisN sT N, whereS= diag (1,−1, . . . ,(−1)ⁿ⁺¹).

Lemma 1.4. [FJ11, Jacobi’s Identity, pp. 28-29] Let A ∈ R^n,n be nonsingular. Then for any nonempty subsets α, β⊆ {1, . . . , n} with|α|=|β|the following equality holds:

detA⁻¹[α|β] = (−1)^sdetA[β^c|α^c]

detA , (1.7)

where s:=^P_α

i∈ααi+^P_β

i∈ββi.

The next lemma relates the determinant of a given square matrix to its minors.

Lemma 1.5. [Kar68, Laplace Expansion by Minors, p. 6] Let A ∈ R^n,n. Then for any k= 1, . . . , n and a fixed α∈Q_k,n we have

detA= ^X

β∈Qk,n

(−1)^sdetA[α|β] detA[α^c|β^c], where sis defined for each α, β ∈Q_k,n as in the above lemma.

The next lemma will be applied to recognize SSR matrices.

Lemma 1.6. [GK02, Lemma 1, p. 259] For an arbitrary matrix A = (a_ij) ∈R^n+1,n, the following equality holds:

detA[1, . . . , n|1, . . . , n] detA[2, . . . , n−1, n+ 1|1, . . . , n−1] +

detA[n,2, . . . , n−1, n+ 1|1, . . . , n] detA[2, . . . , n−1,1|1, . . . , n−1] + detA[n+ 1,2, . . . , n−1,1|1, . . . , n] detA[2, . . . , n−1, n|1, . . . , n−1] = 0.

(21)

The following lemma provides a determinantal identity which can be viewed as a generalization of [And87, (1.39)], see also [GLL11b, (B. 1)].

Lemma 1.7. Let A ∈ R^n,m, α = (α₁, . . . , α_l) ∈ Q_l,n, and β = (β₁, . . . , βl−1) ∈ Ql−1,m−1

with 0< d(β). Then for all η such that βl−1 < η ≤m, k ∈ {1, . . . , l}, s∈ {1, . . . , h}, and β_h < t < β_h+1 for some h ∈ {1, . . . , l−2} or βl−1 < t < η the following determinantal identity holds:

detA[ααˆk|β_β_ˆ

s∪ {t}] detA[α|β∪ {η}] = detA[ααˆk|β_β_ˆ

s∪ {η}] detA[α|β∪ {t}]

+ detA[α_α_ˆ_k|β] detA[α|β_β_ˆ

s ∪ {t, η}]. (1.8) Proof. The proof proceeds by applying Lemma 1.3 to the following (l+ 1)-by-(l+ 1) matrix

B :=







a_α₁_,β₁ . . . a_α₁_,β_h a_α₁_,t a_α₁_,β_h+1 . . . a_α₁_,β_l−1 a_α₁_,η aα2,β1 . . . aα2,βh aα2,t aα2,βh+1 . . . aα2,βl−1 aα2,η

... . . . ... ... ... . . . ...

aα_l,β1 . . . aα_l,β_h aαl,t aα_l,β_h+1 . . . aα_l,βl−1 aαl,η

0 . . . 0 1 0 . . . 0 0







. (1.9)

The determinants of some special kind of matrices can be evaluated by using recursion relations, for instance, the determinant of an n-by-n tridiagonal matrix A = (a_ij) can be evaluated by using the following recursion equations:

detA=a11detA[2, . . . , n]−a12a21detA[3, . . . , n] (1.10)

=anndetA[1, . . . , n−1]−an−1,nan,n−1detA[1, . . . , n−2]. (1.11) The next proposition extends the above two relations.

Proposition 1.1. [Pin10, Formula (4.1)] For an n-by-n tridiagonal matrix A = (a_ij) the following relation holds:

detA= detA[1, . . . , i−1] detA[i, . . . , n] (1.12)

−ai−1,iai,i−1detA[1, . . . , i−2] detA[i+ 1, . . . , n], i= 2, . . . , n.

Now we turn to matrix inequalities that hold for T N matrices.

Theorem 1.3. [GK02, Theorem 16, p. 270] Let A= (aij)∈R^n,n be T N. Then a₁₁detA(1) − a₁₂detA(1|2) +. . .−a_1,2pdetA(1|2p)≤detA

≤ a11detA(1)−a12detA(1|2) +. . .+a1,2q−1detA(1|2q−1)

p= 1,2, . . . , n

2

; q= 1,2, . . . ,

n+ 1 2

.

(22)

Theorem 1.4. [FJ11, Corollary 6.2.4, Koteljanski˘i Inequality] Let A∈R^n,n be T N. Then for anyα∈Q_k,n and β ∈Q_l,n, the following inequality holds:

detA[α∪β]·detA[α∩β]≤detA[α]·detA[β]. (1.13) We conclude this section with a proposition which provides a useful determinantal inequality that plays a fundamental role in the solution of the perturbation problem of T P matrices.

Proposition 1.2. [Ska04, Theorem 4.2] Let α, α⁰, β, β⁰, γ, γ⁰, δ, δ⁰ be subsets of

{1,2, . . . , n}withα∪γ ={1,2, . . . , p}and α⁰∪γ⁰ ={1,2, . . . , p⁰},q =|α∩γ|, q⁰=|α⁰∩γ⁰|, and r:= ¹₂(p−q+p⁰−q⁰). Let η be the unique order preserving map

η: (α\γ)∪(γ\α)→ {1,2, . . . , p−q}, and let η⁰ be the unique order reversing map

η⁰ : (α⁰\γ⁰)∪(γ⁰\α⁰)→ {p−q+ 1, . . . ,2r}.

Define the subsets α⁰⁰ and β⁰⁰ of {1,2, . . . ,2r} by

α⁰⁰:=η(α\γ)∪η⁰(γ⁰\α⁰), β⁰⁰:=η(β\δ)∪η⁰(δ⁰\β⁰).

Then the following two statements are equivalent:

1. For each square T N matrix A of order at least n the following relation holds:

detA[α|α⁰] detA[γ|γ⁰]≤detA[β|β⁰] detA[δ|δ⁰].

2. The relations α∪γ =β∪δ andα⁰∪γ⁰ =β⁰∪δ⁰ are fulfilled and the setsα⁰⁰, β⁰⁰ satisfy the inequality

max{|ω∩β⁰⁰|,|ω\β⁰⁰|} ≤max{|ω∩α⁰⁰|,|ω\α⁰⁰|} (1.14) for each subsetω⊆ {1,2, . . . ,2r} of even cardinality.

(23)

2. Sign Regular, Totally Nonpositive, and Totally Nonnegative Matrices

In order to benefit from the properties of sign regular matrices or of any of its subclasses one needs first to check whether a given matrix is sign regular or not. In this chapter we present several methods in order to check the membership of a given matrix in a certain subclass of sign regular matrices. Furthermore, we present some spectral properties of these subclasses.

These methods consist of determinantal criteria, the variation diminishing property, planar networks, and the Neville elimination. We start with determinantal criteria that are needed in order to check the membership of a given n-by-m matrix in the class of sign regular (totally nonpositive, totally nonnegative) matrices. In the case that we are given a square matrix of ordernand want to employ the definition of sign regularity (totally nonpositivity, totally nonnegativity), we need to check about 4ⁿ(πn)^−1/2minors which is a very large number of minors whenn is relatively large. Thus it is obviously impractical to check whether a matrix is sign regular by naively checking all its minors. So it is very important to search for criteria by which this number of minors could be decreased. In the first three sections we are concerned with the recognition problem: can one decide, by checking a restricted number of minors, whether or not a real matrix is sign regular of any subclass? Many of the inequalities in the definition of a strictly sign regular matrix are superfluous as we see later.

The organization of this chapter is as follows. In Section 2.1, determinantal criteria for checking sign regularity, strict sign regularity, and almost strict sign regularity, and related results are given. In Section 2.2, we focus on a subclass of sign regular matrices which is the class of the totally nonpositive matrices and present some determinantal criteria and properties of this subclass. In Section 2.3, we turn to another important subclass of sign regular matrices which is the class of the totally nonnegative matrices and introduce some determinantal criteria and properties of it and its subclasses. In Section 2.4, we present the variation diminishing property and its connections to the sign regular matrices. In Section 2.5, the relationships between totally nonnegative matrices and planar networks are investigated. In Section 2.6, the Neville elimination is introduced and its usefulness in ascertaining the total nonnegativity of a given matrix is discussed. Finally, in Section 2.7, some spectral properties of totally positive, oscillatory, totally nonnegative, totally negative, and totally nonpositive matrices are given.

(24)

2.1. Sign Regular Matrices

In this section we present some determinantal conditions that are sufficient for a given matrix to be strictly sign regular or sign regular. We start with strictly sign regular matrices and introduce an efficient determinantal criterion for checking strictly sign regularity. The following theorem states that it is sufficient to check only contiguous minors for strictly sign regularity. We remind the reader that we have defined in Section 1.2n⁰= min{n, m}.

Theorem 2.1. [GK02, Corollary, p. 261], [And87, Theorem 2.5] Let A∈R^n,m. Then A is SSR with signature = (₁, . . . , _n⁰) if

0< _kdetA[α|β] wheneverα∈Q_k,n, β ∈Q_k,m, and d(α) =d(β) = 0, k = 1, . . . , n⁰.(2.1) Theorem 2.1 in the case ofT P matrices is proved by Fekete [Fek13]. The following theorem shows that the rank of a given matrix plays a significant role in reducing the number of minors that one needs to check for sign regularity. The case ofT N matrices was proved in [Cry76].

Theorem 2.2. [And87, Theorem 2.1] Let A∈R^n,m be of rank r and = (₁, . . . , _n⁰) be a signature sequence. If

0≤_kdetA[α|β] for α∈Q_k,n, β∈Q_k,m, k= 1,2, . . . , r, (2.2) is valid whenever d(β)≤m−r, then A isSR with signature .

As a consequence of Theorem 2.2, if a given square matrix is nonsingular then it is sufficient to check the minors corresponding to submatrices whose columns are consecutive.

Unfortunately, this is all what can be positively said concerning the minimal number of minors that is sufficient for sign regularity of arbitrary signature. The following theorem can be easily concluded by using the Cauchy-Binet Identity; it shows that the set of sign regular matrices is closed under matrix multiplication.

Theorem 2.3. [And87, Theorem 3.1]LetA, B∈R^n,nbeSRwith signatures= (1, . . . , n) and δ= (δ₁, . . . , δ_n), respectively. Then AB isSR with signature (₁δ₁, . . . , _nδ_n).

In the next subsection and two sections we present some subclasses ofSRmatrices which need a fewer number of minors to be checked than that of generalSR matrices.

There are no efficient criteria for general SR matrices as forSSR matrices but each SR matrix can be approximated arbitrary closely by anSSRmatrix. The next theorem, which was first showed by A. M. Whitney [Whi52] in the case ofT N matrices, shows that we can approximate any square SR matrix by an SSR and for a given square SR matrix whose rank is less than its order, we can approximate it by a strictly one not only with the same signature but also with a possible different signature.

Theorem 2.4. [GK02, Theorem 17, p. 272], [And87, Theorem 2.7] Every SR matrix A∈R^n,n can be approximated arbitrarily closely by SSR matrices with the same signature.

Moreover, if A has rank r, r < n, then it can be approximated by a SSR matrix with arbitrary prescribed signs of the minors of order greater than r.

(25)

Several decompositions ofSSRmatrices are studied in [CPn08]. In [BPn12] it was shown that a given tridiagonal SR matrix can have only one of the following special signature sequences:

= (1,1, . . . ,1, _n), or (2.3)

= (−1,1, . . . ,(−1)ⁿ⁻¹, n), (2.4) withn=±1.

The next two theorems provide efficient criteria for sign regularity of a given nonnegative tridiagonal matrix.

Theorem 2.5. [HC10, Theorem 9]Let A∈R^n,nbe a nonnegative tridiagonal matrix. Then the following three statements are equivalent:

(a) A isN sSR with signature = (1,1, . . . ,1, n).

(b) The following conditions are satisfied:

0< _ndetA, (2.5)

0≤detA[2, . . . , n], (2.6)

0≤detA[1, . . . , n−1], (2.7)

0<detA[1, . . . , k], for all1≤k≤n−2. (2.8) (c) The following conditions are satisfied:

0< _ndetA, (2.9)

0≤detA[2, . . . , n], (2.10)

0≤detA[1, . . . , n−1], (2.11)

0<detA[k+ 1, . . . , n], for all2≤k≤n−1. (2.12) Theorem 2.6. [BPn12, Theorem 4.1]Let3≤n, andA∈R^n,nbe a nonsingular nonnegative tridiagonal matrix. Then A is SR if and only if A[1, . . . , n−1] and A[2, . . . , n] are T N, and A[1, . . . , n−2]and A[2, . . . , n−1] are nonsingular.

We close this section with the following subsection which introduces an intermediate subclass between the SR andSSR matrices.

2.1.1. Almost Strictly Sign Regular Matrices

In this subsection we consider a subclass of sign regular matrices that is intermediate between the sign regular and the strictly sign regular matrices.

We start with the following lemma which characterizes the zero-nonzero pattern of a given N sSRmatrix according to the value of the second component of its signature.

(26)

Lemma 2.1. [HLZ12, Lemma 7], see also [Pn02, Lemma 2.2] Let A = (aij) ∈ R^n,n be N sSR with signature = (₁, . . . , _n). Then one of the following two statements holds:

(i) If 2 = 1, then







aii6= 0, i= 1, . . . , n,

aij = 0, j < i ⇒ a_kl= 0 ∀l≤j < i≤k, a_ij = 0, i < j ⇒ a_kl= 0 ∀k≤i < j≤l, which is called atype-I staircase matrix.

(ii) If ₂ =−1, then







a_1n6= 0, a2,n−1 6= 0, . . . , a_n1 6= 0,

aij = 0, n−i+ 1< j ⇒ akl= 0 ∀i≤k, j≤l, a_ij = 0, j < n−i+ 1 ⇒ a_kl= 0 ∀k≤i, l≤j, which is called atype-II staircase matrix.

Following [HLZ12], we call a minor trivial if it vanishes and its zero value is determined already by the pattern of its zero-nonzero entries. We illustrate this definition by the following example. Let

A:=







∗ ∗ ∗ 0 ∗ 0 0 ∗ ∗





,

where the asterisk denotes a nonzero entry. Then detA[2,3|1,2] and detA[1,2|1,3] are trivial, whereas detA and detA[1,2|2,3] are nontrivial minors.

The following lemma identifies the trivial/nontrivial minors in a given staircase matrix.

Lemma 2.2. [HLZ12, p. 4183] Let A= (aij) ∈R^n,n and α, β ∈ Qκ,n. Then the staircase matrices possess the following properties:

(i) If A is a type-I staircase matrix, then

detA[α|β]is a nontrivial minor ⇔ a_α₁_,β₁·a_α₂_,β₂· · ·a_α_κ_,β_κ 6= 0.

(ii) If A is a type-II staircase matrix, then

detA[α|β]is a nontrivial minor ⇔ aα1,βκ·aα2,βκ−1· · ·aακ,β1 6= 0.

(iii) A is a type-I staircase matrix if and only ifT_nA is a type-II staircase matrix, and detA[α|β]is a nontrivial minor ⇔ det (T_nA)[α⁰|β]is a nontrivial minor, where α⁰ := (n−αi+ 1, i= 1, . . . , κ).

(27)

Now we present the definition of the almost strictly sign regular matrices and characterize them.

Definition 2.1. [HLZ12, Definition 8] Let A ∈ R^n,n and = (₁, . . . , _n) be a signature sequence. If for all the nontrivial minors the following inequalities

0< _kdetA[α|β], where α, β∈Q_k,n, k = 1, . . . , n,

hold, thenA is called almost strictly sign regular(abbreviated ASSR)with signature . The next theorem provides an efficient criterion for nonsingular almost strictly sign regularity.

Theorem 2.7. [HLZ12, Theorem 10] Let A ∈ R^n,n and = (1, . . . , n) be a signature sequence. Then A is N sASSR with signature if and only if A is a type-I or type-II staircase matrix, and for all the nontrivial minors the following inequalities hold:

0< _kdetA[α|β], where α, β∈Q_k,n such that d(α) =d(β) = 0, k= 1, . . . , n.

One can predict some components of the signature sequence of a given type-I staircase matrix by using the positions of vanishing entries of a givenN sASSR matrix.

Lemma 2.3. [HLZ12, Lemma 9] Let A = (aij) ∈ R^n,n be a type-I staircase matrix and N sASSR with signature = (₁, . . . , _n). Set

r:= min{|j−i| |aij = 0, for some i, j∈ {1, . . . , n}}. Then the following relations hold

₂=²₁, ₃=³₁, . . . , n−r+1 =^n−r+1₁ .

In [APnS15],N sASSR matrices are studied and characterized through the Neville elimination, see Section 2.6.

2.2. Totally Nonpositive Matrices

In this section we present a subclass of sign regular matrices which is called totally nonpositive matrices, some of its properties, and determinantal criteria for checking the membership of a given matrix in this subclass.

Recall that a given matrix is called totally nonpositive t.n.p. (totally negative t.n.) if all of its minors are nonpositive (negative). The study oft.n.matrices was initiated in [FD00].

Some aspects of this subclass including existence, spectral properties, Schur complements, and factorizations were considered therein. The following propositions introduce methods to generateT N,T P,t.n.p., and t.n.matrices from givenT P,t.n., and t.n.p.matrices.

Proposition 2.1. [FD00, Proposition 2.1]LetA∈R^n,m bet.n.(t.n.p.) and letB, C ∈R^m,p be t.n. andT P, respectively. Then AB∈R^n,p is T P (T N) and AC ∈R^n,p is t.n.(t.n.p.).