Matrices and Graphs

Theory and Applications

to Economics

Proceedings of the Conferences on

Matrices and Graphs

Theory and Applications

to Economics

University of Brescia, Italy

8 June 1993

22 June 1995

Sergio Camiz

Dipartimento di Matematica "Guido Castelnuovo"

Universita di Roma "La Sapienza", Italy

Silvana Stefani

Dipartimento Metodi Quantitativi

Universita di Brescia, Italy

b World Scientific

II

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MATRICES AND GRAPHS

Theory and Applications to Economics

Copyright © 1996 by World Scientific Publishing Co. Pte. Ltd.

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v

FOREWORD

the editors

The idea to publish this book was born during the conference « Matrices

and Graphs: Computational Problems and Economic Applications» held in

the far June 1993 at Brescia University. The conference was such a success that

the organizers, actually the present editors themselves, after a short talk with

the lecturers, decided on the spot to apply to the Italian Consiglio Nazionale

delle Ricerche (CNR) for a contribution to publish the conference proceedings.

The second editor did and the contribution came after a while.

In the meantime the editors organized another conference on «Matrices

and Graphs: Theory and Economic Applications», held as the previous in

Brescia during June 1995, partly with different invited lecturers. The conference was a success again and therefore the first editor applied to the Italian

National Research Council and got a second contribution, that came only recently.

While the lecturers of the first conference, who were not at the second one,

were a bit upset, having submitted their paper without seeing any proceedings

published at that time, the lecturers common to the first and the second conference suggested to join the contributions and publish a unique book for both

conferences. This is what we did.

During all these years, both authors were very busy lecturing, researching,

publishing, raising more funds to make their research possible. Most papers

arrived late and were carefully read by the editors, then the search of suitable

referees was not easy, so that the reviewing process took also a while, some

papers being sent back to the authors for corrections and then submitted again

to referees. A complete re-editing was necessary in order to get the uniform

editor's style ... , well, these are the reasons of such a delay, but eventually here

we are.

The book reflects our scientific research background: for academic and scientific reasons both of us were drawn to different research subjects, both shifting from pure to applied mathematics and statistics, with particular attention

to data analysis in many different fields the first editor, and to operational

research and mathematical finance the second. So, in each of the steps of this

long way, we collected a bit of knowledge.

The fact that in most of investigations we dealt with matrices and graphs

suggested us to investigate in how many different situations they may be used.

vi

This was the reason that led to the conferences; as a result, this book looks

like a patchwork, as it is composed of different aspects: we submit it to the

readers, hoping that it will be appreciated, as we did.

In fact, the numerous contributions come from pure and applied mathematics, operations research, statistics, econometrics. Roughly speaking, we

can divide the contributions by areas: Graphs and Matrices, from theoretical

results to numerical analysis, Graphs and Econometrics, Graphs and Theoretical and Applied Statistics.

Graphs and Matrices contributions begin with John Maybee: in his paper New Insights on the Sign Stability Theorem, he finds a new characterization

of a sign stable matrix, based on some properties of the eigenvectors associated to a sign semi-stable matrix. Szolt Thza in Lower Bounds for a Class

of Depth- Two Switching Circuits obtains a lower bound for a certain class of

(0,1) matrices. It is interesting to note that the problem can be formulated

in terms of a semicomplete digraph D, if one wants to determine the smallest

sum of the number of vertices in complete bipartite digraphs, whose union is

the digraph D itself. Tiziana Calamoneri and Rossella Petreschi's Cubic

Graphs as Model of Real Systems is a survey on cubic graphs, i.e. regular

graphs of degree three, and at most cubic graphs, i.e. graphs with maximum

degree three and show a few applications in probability, military problems, and

financial networks. Silvana Stefani and Anna Torriero in Spectral Properties of Matrices and Graphs describe from one hand how to deduce properties

of graphs through the spectral structure of the associated matrices and on the

other how to get information on the spectral structure of a matrix through

associated graphs. New results are obtained towards the characterization of

real spectrum matrices, based on the properties of the associated digraphs.

Guido Ceccarossi in Irreducible Matrices and Primitivity Index obtains a

new upper bound for the primitivity index of a matrix through graph theory

and extends this concept to the class of periodic matrices. Sergio Camiz and

Yanda Thlli in Computing Eigenvalues and Eigenvectors of a Symmetric Matrix: a Comparison of Algorithms compare Divide et Impera, a new numerical

method for computing eigenvalues and eigenvectors of a symmetric matrix, to

more classical procedures. Divide et Impera is used to integrate those procedures based on similarity transformations at the step in which the eigensystem

of a tridiagonal matrix has to be computed.

Among contributions on Graphs and Econometrics we find Sergio Camiz

paper I/O Analysis: Old and New Analysis Techniques. In this paper, Camiz

compares various techniques used in I/O analysis to reveal the complex struc-

vii

ture of linkages among economic sectors: triangularization, linkages comparison, exploratory correspondence analysis, etc. Graph analysis, with such concepts as centrality, connectivity, vulnerability, turns out to be a useful tool for

identifying the main economic flows, since it is able to reveal the most important information contained in the I/O table. Manfred GilU in Graphs and

Macroeconometric Modelling deals with the search of a local unique solution of

a system of equations and with necessary and sufficient conditions for this s0lution to hold. He shows how through a graph theoretic approach the problem

can be efficiently investigated, in particular when the Jacobian matrix is large

and sparse, a typical case of most econometric models. Manfred Gilli and

Giorgio Pauletto in Qualitative Sensitivity Analysis in Multiequation Models

perform a sensitivity analysis of a given model when a linear approximation is

used, the sign is given and there are restrictions on the parameters. They show

that a qualitative approach, based on graph theory, can be fruitful and lead

to conclusions which are more general than the quantitative ones, as they are

not limited to a neighborhood of the particular simulation path used. Mario

FaUva in Hadamard Matrix Product, Graph and System Theories: Motivations

and Role in Econometrics shows how the analysis of a model's causal structure can be handled by using Hadamard product algebra, together with graph

theory and system theoretical arguments. As a result, efficient mathematical

tools are developed, to reveal the causal and interdependent mechanisms associated with large econometric models. At last, International Comparisons and

Construction of Optimal Graphs, by Bianca Maria Zavanella, contains an

application of graph theory to the analysis of the European Union countries

based on prices, quantities and volumes. Graph theory turns out to be a most

powerful tool to show which nations are more similar.

Graphs and Statistics papers are represented by three contributions. Giovanna lona Lasinio and Paola Vicard in Graphical Gaussian Models and

Regression review the use of graphs in statistical modelling. The relative merits of regression and graphical modelling approach are described and compared

both form the theoretical point of view and with application to real data.

Francesco Lagona in Linear Structural Dependence of Degree One among

Data: a Statistical Model models the presence of some latent observations using

a linear structural dependence among data, thus deriving a particular Markovian Gaussian field. Bellacicco and Tulli in Cluster identification in a signed

graph by eigenvalue analysis establish a connection between clustering analysis

and graphs, by including clustering into the wide class of a graph transformation in terms of cuts and insertion of arcs to obtain a given topology.

viii

After this review, it should be clear how important is the role of matrices

and graphs and their mutual relations, in theoretical and applied disciplines.

We hope that this book will give a contribution to this understanding.

We thank all the authors for their patience in revising their work. A

special thanks goes to Anna Torriero and Guido Ceccarossi for their constant

help, but especially we would like to thank Yanda Tulli, who did the complete

editing trying to (and succeeding in) making order among the many versions

of the papers we got during the revision process. Last, but not least, thanks

to Mrs. Chionh of World Scientific Publishers in Singapore, whom we do not

know personally, but whose efficieny we had the opportunity to know through

E-mail.

October, 1996.

Sergio Camiz and Silvana Stefani

The manuscripts by Sergio Camiz, Guido Ceccarossi, Manfred Gilli, Giovanna lona Lasinio and Paola Vicard, Francesco Lagona, and Bianca Maria

Zavanella, referring to the first Conference, have been received at the end of

1993. The manuscripts of Antonio Bellacicco and Yanda Tulli, Tiziana Calamoneri and Rossella Petreschi, Sergio Camiz and Yanda Tulli, Mario Faliva,

Manfred Gilli and Giorgio Pauletto, John Maybee, Silvana Stefani and Anna

Torriero, and Szolt Tuza, referring to the second Conference, were received at

the end of 1995.

This work was granted by the contributions from Consiglio Nazionale delle

Ricerche n. A.I. 94.00967 (Silvana Stefani) and Consiglio Nazionale delle

Ricerche n. A.I. 96.00685 (Sergio Camiz).

ix

Sergio Camiz is professor of Mathematics at the Faculty of Architecture of

Rome University ~La Sapienza~. In the past, he was professor of Mathematics at Universities of Calabria, Sassari, and Molise, of Statistics at Benevento

Faculty of Economics of Salerno University, and of Computer Science at the

American University of Rome. He spent periods as visiting professor at the

Universities of Budapest (Hungary), Western Ontario (Canada), Lille (France)

and at Tampere Peace Research Institute of Tampere University (Finland),

contributed to short courses on numerical ecology in the Universities of Rome,

Rosario (Argentina), and Leon (Spain), held conferences on data analysis applications at various italian universities, as well as the Universities of New

Mexico (Las Cruces), Brussels (Belgium), Turku and Tampere (Finland), and

at IADIZA in Mendoza (Argentina), contributed with communications to various academic congresses in Italy, Europe, and America. After a long activity in

the frame of computational statistics and data analysis for numerical ecology,

and in programming numerical computations in econometrics and in applied

mechanics, his present research topics concern the analysis, development, and

use of numerical mathematical methods for data analysis and applications in

different frames, such as economical geography, archaeology, sociology, and

political sciences. He was co-editor of two books, one concerning the analysis

of urban supplies and the other on pollution problems, and author of several

papers published on scientific journals.

Silvana Stefani is a Full Professor of Mathematics for Economics at the

University of Brescia. She got her Laurea in Operations Research at the University of Milano. She has been visiting scholar in various Universities in Warsaw (Poland), Philadelphia (USA), Jerusalem (Israel), Rotterdam (the Netherlands), New York and Chicago (USA). She has been Head of the Department of

Quantitative Methods, University of Brescia, from November 1990 to October

1994 and is currently Coordinator of the Ph.D. Programme ~Mathematics

for the Analysis of Financial Markets~. She was co-editor of two books, one

concerning the analysis of urban supplies and the other on mathematical methods for economics and finance, and author of numerous articles, published in

international Journals, in Operations Research, applied Mathematics, Mathematical Finance.

Typeset by Jj.TEX

Edited by Vanda Tulli

x

AUTHORS ADDRESSES

Antonio Bellacicco Dipartimento di Teoria dei Sistemi e delle Organizzazioni

Universita di Teramo

Via Cruccioli, 125, 64100 Teramo, Italia.

Tiziana Calamoneri Dipartimento di Scienza dell'Informazione

Universita "La Sapienza"di Roma

Via Salaria, 113, 00198 Rama, Italia.

E-mail: Calamo@dsi.uniroma1.it

Sergio Camiz Dipartimento di Matematica "Guido Castelnuovo"

Universita "La Sapienza"di Roma

P.le A. Moro, 2, 00185 Roma, Italia.

E-mail Camiz@mat.uniroma.it

Luigi Guido Ceccarossi Dipartimento Metodi Quantitativi

Universita di Brescia

Contrada S. Chiara 48/b, 25122 Brescia, Italia.

E-mail Ceccaros@master.cci.unibs.it

Mario Faliva Istituto di Econometria e Matematica per Ie Decisioni Economiche

Universita Cattolica di Milano

Via Necchi 9, 20100 Milano, Italia.

Manfred Gilli Departement d':Econometrie

U niversite de Geneve

Boulevard Carl-Vogt 102, 1211 Geneve 4, Switzerland.

E-mail Manfred.Gilli@metri.unige.ch

Giovanna lona Lasinio Dipartimento di Statistica, Probabilita e Statistiche

Applicate

Universita "La Sapienza"di Roma

P.le A. Moro, 2, 00185 Rama, Italia.

E-mail Iona@pow2.sta.uniroma.it

Francesco Lagona Dipartimento di Statistica, Probabilita e Statistiche Applicate

Universita "La Sapienza"di Roma

P.le A. Moro, 2, 00185 Rama, Italia.

xi

John Maybee University of Colorado

265 Hopi PI. Boulder, Co 80303 USA.

E-mail Maybee@newton.colorado.edu

Giorgio Pauletto Departement d'Econometrie

Universite de Geneve

Boulevard Carl-Vogt 102, 1211 Geneve 4, Switzerland.

E-mail Giorgio.Pauletto@metri.unige.ch

Rossella Petreschi Dipartimento di Scienza dell'Informazione

Universita "La Sapienza di Roma "

via Salaria, 113, 00198 Roma, Italia.

E-mail: Petreschi@dsi.uniroma1.it

Silvana Stefani Dipartimento Metodi Quantitativi

Universita di Brescia

Contrada S. Chiara 48/b, 25122 Brescia, Italia.

E-mail Stefani@master.cci.unibs.it

Anna Torriero Istituto di Econometria e Matematica per Ie Decisioni Economiche

U niversita Cattolica di Milano

Via Necchi 9, 20100 Milano, Italia.

E-mail Torriero@aixmiced.mi.unicatt.it

Vanda Tulli Dipartimento Metodi Quantitativi

Universita di Brescia

Contrada S. Chiara 48/b, 25122 Brescia, Italia.

Zsolt Tuza Computer and Automation Research Institute

Hungarian Academy of Sciences

1111 Budapest, Kende u. 13-17, Hungary.

e-mail tuza@lutra.sztaki.hu

Paola Vicard Dipartimento di Statistica, Probabilita e Statistiche Applicate

Universita "La Sapienza"di Roma

P.le A. Moro, 2, 00185 Roma, Italia.

Bianca Maria Zavanella Istituto di Statistica, Facolta di Scienze Politiche

Universita Statale di Milano

Via Visconti di Modrone, 20100 Milano, Italia.

E-mail Zavanell@imiucca.unimi.it

xiii

Contents

New Insights on the Sign Stability Problem

John Maybee

1

Lower Bounds for a Class of Depth-two Switching Circuits

ZsoltTuza

7

Cubic Graphs as Model of Real Systems

Tiziana Calamoneri and Rossella Petreschi

19

Spectral Properties of Matrices and Graphs

Silvana Stefani and Anna Torriero

31

Irreducible Matrices and Primitivity Index

Luigi Guido Ceccarossi

50

A Comparison of Algorithms for Computing the Eigenvalues

and the Eigenvectors of Symmetrical Matrices

Sergio Camiz and Yanda Tulli

I/O Analysis: Old and New Analysis Techniques

Sergio Camiz

72

92

Graphs and Macroeconometric Modelling

Manfred Gilli

120

Qualitative Sensitivity Analysis in Multiequation Models

Manfred Gilli and Giorgio Pauletto

137

Hadamard Matrix Product, Graph and System Theories: Motivations

and Role in Econometrics

Mario Faliva

International Comparisons and Construction of Optimal Graphs

Bianca Maria Zavanella

152

176

xiv

Graphical Gaussian Models and Regression

Giovanna Jona Lasinio and Paola Vicard

Linear Structural Dependence of Degree One among Data: A

Statistical Model

Francesco Lagona

Cluster Identification in a Signed Graph by Eigenvalue Analysis

Antonio Bellacicco and Yanda Tulli

200

223

233

NEW INSIGHTS ON THE SIGN STABILITY PROBLEM

J. MAYBEE

Progmm in Applied Mathematics

University of Colomdo

We obtain a new characterization of when a matrix is sign stable. Our results

makes use of properties of eigenvectors of sign semi-stable matrices. No classical

stability theorems are required in proving our results.

1

Introduction

We deal with n x n real matrices. Such a Matrix A is called semistable (stable)

if every A in the spectrum of A, u(A) lies in the closed (open) left-half of the

complex plane. The real matrix sgn(A) = [sgn aij] is called the sign pattern of

A and two real matrices A and B are said to have the same sign pattern if either

aij bij > 0 or both aij and bij are zero for all i and j. When A is a real matrix

we let Q(A).be the set of all matrices having the same sign pattern as A. We

also will write A in the form A = Ad + A where Ad = diag[an, a22, ... ,ann]

and A= A-Ad.

Let u be a complex vector u = (Ul' U2, ... , un). We say that U is qorthogonal to Ad if au # 0 implies Ui = O. Notice that if U is q-orthogonal to

Ad, then U is q-orthogonal to Bd for every matrix BE Q(A).

Let A be a real matrix satisfying aij # 0 if and only if aji # 0 for all i # j.

Then A is called combinatorially symmetric and we may associate with A the

graph G(A) having n vertices and an edge joining vertices i and j if and only if

i # j and aij # o. The graph G(A) is a tree if it is connected and acyclic. We

also use, for any matrix, the directed graph D(A) defined in the usual way.

The real matrix A is called sign semi-stable (sign-stable) if every matrix

in Q(A) is semi-stable (stable). We will deal only with the case where A is

irreducible in order to keep the arguments simple (Gantmacher, 1964). All of

our results can be readily extended to the reducible case.

We will prove the following results about sign semi-stable matrices.

Theorem 1 The following are equivalent statements:

1. The matrix A is sign semi-stable.

2. Matrix A satisfies

(i) ajj S; 0, for all j,

2

(ii) aijaji ::; 0 for all i and j, and

(iii) every product of the form ai(1)i(2)ai(2)i(3) ... ai(k)i(l) = 0 for k 2:

3 where {i(I), i(2), ... , i(k)} is a set of distinct integers in N =

{l,2, ... ,n}.

3. There exists a positive diagonal matrix D = diag[dl, d2 , ... , dnl, di >

0, i = 1,2, ... , n such that the matrix DAD- 1 = Ad + S where S is skew

symmetric (Gantmacher, 1964) and satisfies (ii), and (iii).

Theorem 2 The following are equivalent statements about a sign semi-stable

matrix:

I'. The matrix A has .A = 0 as an eigenvalue.

2'. Every matrix in Q(A) has .A

=0

as an eigenvalue.

3'. There is an eigenvector u satisfying Au = 0 which is q-orthogonal to Ad.

Theorem 3 The following are equivalent statements about a sign semi-stable

matrix:

I". The matrix A does not have a purely imaginary eigenvalue.

2". No matrix in Q(A) has a nonzero purely imaginary eigenvalue.

3". There is no eigenvector u satisfying Au = ij.Lu, which is q-orthogonal to

Ad.

The equivalence of conditions (1) and (2) of Theorem 1 is a well known

result due to Maybee, Quirk, and Ruppert (see Jefferies et al, 1977 for one

proof of this result). All the known proofs of this equivalence make use of one

of the classical stability theorems. By proving that (1) ::::} (2) ::::} (3) ::::} (1) we

can avoid the use of any stability theorem, a fact of some independent interest.

A consequence of Theorem 1 is that the family of sign semi-stable matrices

can be identified with the family of matrices of the form A = Ad + S, where

S is skew-symmetric and A satisfies (i),(ii), and (iii). This fact is used in an

essential way to prove Theorem 3.

Our proofs of Theorems 2 and 3 lead directly to simple algorithms for

testing a given matrix satisfying the conclusions (i),(ii), and (iii) to determine

whether or not it is sign-stable.

Finally, given Theorems 1,2, and 3 we can state the following sign stability

result.

3

Theorem 4 The real matrix A is sign stable if and only if the following four

conditions are satisfied.

(i) ajj ::; 0 for all j;

(ii) aijaji ::; 0 for all i and j;

(iii) every product of the form ai(1)i(2)' ai(2)i(3) ... ai(k)i(l) = 0 for k 2: 3

where {i(l), i(2), ... ,i(k)} is a set of distinct integers in N = {I, 2, ... ,n}

(iv) the matrix A does not have an eigenvector q-orthogonal to Ad

2

The proof of Theorem 1

Suppose first that the matrix A is sign semi-stable. The fact that (i),(ii),

(iii), are then true follows by a familiar continuity argument which we omit.

Hence (1) ::::} (2). Given that (2) is true and A is irreducible it follows that,

if aij :f 0 then aji :f 0 also. For suppose aij :f 0 and aji = O. Since there

is a path from j to i in A, (iii) is violated. Thus A is combinatorially symmetric. But then G(A), the graph of A exists, is connected and has no cycles. Hence G(A) is a tree. Then by a theorem of Parter and Youngs (1962),

there exists a positive diagonal matrix D such that DAD- 1 = Ad + S where

Ad = diag[all' a22, ... , ann], S = [Sij], with Sii = 0 for i = 1,2, ... , n, and

Sij = -Sji for all i :f j. Thus (2) ::::} (3). Now set A = DAD-l and suppose Au = AU. Taking scalar products on the right and left with U yields

u·Au = U· Adu+u· Su = U· AU = ~lul2 and Au·u = Adu·u+Su·u = Alul 2.

We have U· AdU = Adu· u and U· Su = -Su· u. Hence 2Adu· u = (A + ~)luI2

so we obtain

Re(A) =

AI~~ u

L aiilu l2 where Io =

(1)

{j I ajj :f O}. Hence condition (i)

iE10

implies that for any A in a(A), Re(A) ::; O. Thus (3) implies (1) and Theorem

1 is proved.

Now a sign semi-stable matrix is sign stable if and only if it has no eigenvalues on the imaginary axis in the complex plane. On the other hand, if u is

an eigenvector of A belonging to an eigenvalue on the imaginary axis, then we

must have Ui = 0 if i E Io by (1), i.e. u is q-orthogonal to Ad.

Note also that it follows from the proof of Theorem 1 that, if aii = 0, i =

1 ... n, A is skew-symmetric and all the eigenvalues of A are purely imaginary,

hence A is not sign stable. If aii < 0, i = 1 ... n, then every nonzero vector

But AdU . u =

4

satisfies Re()..) < 0 so A is sign stable. Hence the interesting case for sign

stability is 1 :5 1101 < n, which we assume to hold henceforth.

U

3

The proof of Theorem 2

Let A be a sign semi-stable matrix. By conditions (i) and (ii) every term in

the expansion of det A has the same sign. Therefore if det A = 0, it must

be combinatorially equal to zero and hence every matrix in Q(A) also has

determinant equal to zero. It follows that (1') implies (2'). Our task is to

discover when there exists a non-zero vector U such that Au = O. Now U must

vanish on the (nonempty) set 10 so we partition the components of a candidate

vector U initially into the sets Z(Io), N(Io) where Z(Io) = {i liE I o}, Ui = 0

if i E Z(Io), and Ui =J 0 if i E N(Io). Now given a set Ip ;2 10 and a partition

of the components of U such that Ui = 0 if i E Z(Ip) and Ui =J 0 if i E N(Ip).

We look at the equations

L

SijUj

(2)

= 0

jEN{I,,)

If such an equation has exactly one nonzero term, it has the form SikUk = 0

for some fixed value of k. Since Sik =J 0 and k E N(Ip), this is a contradiction.

Hence we must place k E I p +1 ' We do this for each such occurrence. Thus

IpH ;2 Ip and Z(Ip) ~ Z(Ip+l) , N(Ip) ;2 N(IpH)' If the system (2) contains

no equation having only a single non-zero term, then IpH = Ip and Z(IpH) =

Z(Ip) , N(Ip+l) = N(Ip). We will examine this case below. Suppose that

Ip+l = N. Then Z(IpH) = Nand u=O, i.e. no matrix in Q(A) has zero as

an eigenvalue. It remains to consider the case where we have some Ip = IpH

with IIpl < n so Ui = 0 for i E Z(Ip) and Ui =J 0 for i E N(Ip). Clearly

every equation in system (2) at this point contains either no non-zero terms

or at least two non-zero terms. We have N(Ip) ~ 2 and the induced graph

(N(Ip)) is a forest. We claim that this forest consists of isolated single trees, i.e.

S(N(Ip)) = O. For suppose (N(Ip)) has a nontrivial tree To. This tree has a

vertex of degree one and there would then exist an equation in the subsystem

S(N(Ip))u = 0 having exactly one nonzero term, a contradiction. Next let

IN(Ip) I = q and suppose there exists r rows in the subsystem

SijUi =

L

uEN{Ip)

0, i E Z(Ip), having two or more non-zero entries. We have r ~ 1 so the set

of such rows is nonempty. Let this set be Zo(Ip) and consider the submatrix

S(N(Ip))UZo(Ip). The graph of this submatrix is a forest on the q+r vertices.

If IZo (Ip) I ~ q then the numbers of edges in this forest is at least 2r ~ r + q, a

contradiction. Similarly, there cannot be two directed paths from vertex k to

5

vertex l in the directed graph D(So) where So is the matrix of the subsystem

(2) for i E Zo(Ip). It follows that the subsystem Sou = 0 uniquely determines

one or more eigenvectors U belonging to .A = o. Hence each matrix in Q(A)

has at least one eigenvector U belonging to .A = 0 and vanishing upon the set

Z(Ip) ~ 10. Thus Theorem 2 is proved.

4

The proof of Theorem 3

We look for an eigenvector U such that U is q-orthogonal to Ad and Au = iJL,

for some JL -=I o. As in the proof of Theorem 2, we partition a candidate vector

u initially into the sets Z(Io), N(Io) with Ui = 0 if i E Z(Io) and Ui -=I if

i E N(Io). Now given Ip ;2 10 and a partition of U into Z(Ip) and N(Ip) we

look at the equations

°

L

SjkUk = O,j E Z(Ip),

(3)

kEN(Ip)

and

L

SjkUk = iJLuj,j

E

N(Ip),

(4)

kEN(Ip)

If any equation in the subsystem (3) contains exactly one nonzero term,

we have SjkUk = 0 and, as in the proof of Theorem 2, we adjoin each such k to

Ip. Similarly, if any sum on the left hand side of an equation in the subsystem

(4) contains no nonzero term, we have iJLuj = 0 and this contradiction compels

us to add the index j to Ip. Doing this for every such occurrence produces

the new set IpH ~ Ip and thereby the new partition, Z(IpH), N(IpH). If the

subsystem (3) contains no equation having a single term and the subsystem

(4) contains no empty sums, then Ip+l = Ip. We examine this case below.

Suppose that Ip+l = N. Then Z(IpH ) = Nand U = O. Since every matrix

in Q(A) has the same zero-nonzero pattern, it follows that no matrix in Q(A)

has a purely imaginary nonzero eigenvalue.

It remains to consider the case where we have some Ip = IpH with IIpl < n,

so Ui = 0 for i E Z(Ip) and Ui -=I 0 for i E N(Ip). At this point every equation

in the subsystem (3) has either no nonzero tenns or at least two nonzero terms.

Also the induced subgraph (N(Ip)) is a forest and contains no nontrivial trees,

since every sum in the subsystem (4) contains at least one term. This forest

must contain at least two trees, because if it were a single tree the subsystem

(3) would have an equation containing exactly one nonzero term. Moreover, if

a tree in the forest is adjacent to a vertex j E Z (I p) there must also be another

tree in the forest adjacent to vertex j for the same reason. We must therefore

6

have IN(Ip)1 2:: 4. Let jo be the index of a row in subsystem (3) containing

q 2:: 2 nonzero terms. Then vertex jo in G(A) must be adjacent to q distinct

trees in the forest (N(Ip)).

Now choose a pair of trees in (N(Ip)) adjacent to vertex jo. Set Uk = 0

if k is not a vertex of one of these trees. Let the trees be TI and T2, respectively. Then the submatrices S(TI ) and S(T2) are disjoint nonzero skew

symmetric submatrices of S. Hence they have nonzero purely imaginary eigenvalues iJ.LI and iJ.L2. Let VI and V2 be nonzero vectors satisfying S(TI)VI =

iJ.LI VI, S(T2)V2 = iJ.L2v2. Then any vectors aVI and f3v2 also satisfy these equations where a and f3 are nonzero constants. If J.LI = J.L2 then we choose a and

f3 to satisfy

where kl is a vertex in one tree and k2 is a vertex in the other. Thus aVI and

f3v2 determine a vector u such that Au = iJ.LI u with U q-orthogonal to Ad. If

J.LI #- J.L2 then choose ao such that aoJ.L2 = J.L2 and modify S by multiplying

S(T2) by ao to obtain S'(T2). The resulting matrix is in Q(A) and has iJ.LI as

an eigenvalue.

This proves that some matrix in Q(A) has an eigenvector q-orthogonal

to Ad and it belongs to a purely imaginary eigenvalue. Hence Theorem 3 is

proved.

References

Jefferies C., V. Klee, and P. van den Driessche , 1977.

matrix sign stable?~ Can. J. Math., 29: p. 315-326.

Gantmacher, F. R. , 1964.

New-York.

«When is a

The Theory of Matrices, Vols 1,2. Chelsea,

Parter S. and J. W. T. Youngs, 1962. «The symmetrization of matrices

by diagonal matrices~ J. Math. Anal. Appl., 4: p. 102-110.

7

LOWER BOUNDS FOR A CLASS OF DEPTH-TWO

SWITCHING CIRCUITS

Z. TUZA

Computer and Automation Institute

Hungarian Academy of Sciences

Let M = (aij) be an (m x m) matrix with zero diagonal and ar + a 2i > 0 for

all i =I j, 1 ~ i,j ~ m. For a set R of rows and a set 0 of colutnns, ~enote by

R x 0 the set of the IRI . 101 entries lying in the intersection of those rows and

columns. We prove that if Rl, . .. , Rl and 01, . .. ,Ol are l sets of rows and l sets

l

U(Rk x Ok) is identical to the

of columns of M, respectively, such that the set

set of nonzero entries of M (i.e., aij

=I 0

k=l

if and only if the i-th row is in Rk and

l

the j-th column is in Ok for some k ~ l), then

L

(IRkl

+ lOkI) ~ mlog2 m.

k=l

1

The problem

In this note we investigate an extremal problem on a class of m by m 0-1

matrices, motivated by switching theory. The particular case in question can

be formulated in several equivalent ways, as follows.

• Suppose that the square matrix M = (aij) E {o,l}mxm has zero diagonal, and at least one of aij and aji is 1 for all i f. j, 1 :::; i, j :::; m.

Minimize the total number of rows and columns in a collection of 1-cells

(submatrices with no 0 entry) such that each aij = 1 occurs in at least

one of those I-cells.

• Let B = (X U Y, E) be a bipartite graph with vertex classes X =

{Xl, ... , xm} and Y = {Yl. ... , Ym}, and edge set E, such that XiYi ~ E

for aliI:::; i :::; m, and at least one of XiYj and XjYi belongs to E for all

if. j, 1 :::; i,j :::; m. Find the smallest total number IV(Bl)I+·· .+IV(Be)1

of vertices in a collection of complete bipartite subgraphs Bi C B, Bi =

(Xi U Yi,Ei ), Ei = {xy I X E Xi, Y E Yi} (1 :::; i :::; e), such that

El U···uEe =E.

• Given a semi-complete directed graph D = (V, E) on m vertices, without

loops and parallel edges (i.e., each pair x, Y E V is adjacent either by

just one oriented edge, or by precisely two oppositely oriented edges

8

Xy, YX E E), determine the smallest sum of the numbers of vertices in

complete bipartite digraphs Di C D (with all edges oriented in the same

direction between the two vertex classes in each D i ) whose union is D .

• Suppose that a circuit has to be designed with inputs Xl, . .. ,X m and

outputs YI, . .. ,Ym, where a set of conditions Cij prescribes whether there

exists a directed path of length 2 from Xi to Yj (written as Cij = 1;

otherwise we put Cij = 0). Assuming Cii = 0 for alII::; i ::; m, and

Cij = 1 or Cji = 1 (or both) for all i '" j, 1 ::; i,j ::; m, minimize the

number of links (adjacencies) in such a circuit.

The equivalence of the matrix problem and the two types of graph theoretical formulations is established by the corresponding adjacency matrices:

In the bipartite case we define aij := 1 if and only if Xi is adjacent to Yj; or,

conversely, we join Xi to Yj if and only if aij = 1. For digraphs, the entry

aij = 1 of the matrix corresponds to the edge oriented from vertex i to vertex

j.

To see that the switching circuits also give an equivalent formulation, notice first that each link involved in a path of length 2 verifying Cij = 1 for some

pair i,j either starts from an input node or ends in an output node. Now, each

internal node Zk of a length-2 path connects a set Xk of inputs with a set Yk

of outputs, and the number of links incident to Zk is IXkl + IYkl. Therefore,

X k x Yk must be a I-cell in the 0-1 matrix (Cij). Conversely, each I-cell R xC

of r rows and c columns in a 0-1 matrix M can be represented by an internal

node Z connected to r input nodes and c output nodes in the circuit to be

constructed.

Notation

We denote R xC := {aij I ri E R, Cj E C}, where R ~ {r}, ... ,rm } is a

set of rows and C ~ {CI,"" em} is a set of columns. (We may also view

the 0-1 matrices as subsets of {rl,' .. , rm} X {c}, ... , em}.) The shorthand

l

U (Rk x Ck) =

M means that the entry aij has value 1 in M if and only if

k=l

ri E Rk and

Cj E C k for some k, 1 ::; k ::; i (and aij = 0 otherwise). The

complexity, IT(M), of M is defined as

where the value of i is unrestricted.

9

Obviously, the definition of u(M) can be extended for arbitrary (not necessarily square) 0-1 matrices, but in this paper we do not consider the more

general case; i.e., M E {O, 1}mxm will be assumed throughout.

2

The results

It can be shown (Tuza, 1984) that

cm2

u(M) < logm

holds for every (m x m) matrix M, for some constant c, and also that this

upper bound is best possible apart from the actual value of c. For some restricted classes of matrices, however, the complexity can be much smaller. This

is the case, for example, in the following two particular sequences, as proved

by Tarjan (1975).

Theorem 1 If m = 2n and M = (aij) is the upper triangle matrix (aij = 1

for 1 ::; i < j ::; m and aij = 0 otherwise), then

u(M)

= n . 2n = m log2 m .

/2

Theorem 2 If m = (L n J)' where LxJ is the lower integer part of x, i. e. the

largest integer not exceeding x, and M = (aij) is the matrix J - I with aii = 0

and aij = 1 for all i =I- j, 1::; i,j ::; m, then

Our main goal is to show that the lower bound of m log2 m in Theorem 1 is

valid for a much larger class of (m x m) matrices. Namely, we will prove the

following result:

Theorem 3 If an (m xm) matrix M

and aij + aji > 0 for all i =f:. j, then

= (aij) (aij E {O, I}) has zero diagonal

Theorems 1 and 3 are best possible in general, as it is discussed in the concluding section. On the other hand, we are going to observe that the complexity

10

a(M) of a typical member of the class of matrices involved in Theorem 3 is

much larger than O(mlogm). To formulate this assertion more precisely, denote

E

{O, 1}7nX7n I aii = 0, aij

+ aji > 0

M~ := {M = (aij) E

{O, l}7nX7n I aii = 0, aij

+ aji =

Mm := {M = (aij)

for j

=f i} ,

1 for j

=f i} .

There is a constant c > 0 such that

Theorem 4

cm 2

a(M) 2: - logm

holds for (1 - 0(1)) IMml matrices M E Mm and (1 - 0(1)) IM~I matrices

M E M~ as m ~ 00.

The proofs of Theorems 3 and 4 are given in Sections 3 and 4, respectively.

Some open problems are mentioned in Section 5.

3

The general lower bound

The subject of this section is to prove Theorem 3, i.e., that a(M) 2: m log2 m

holds for all matrices M = (aij) of order m with zero diagonal, containing at

least one nonzero entry in each {aij, aji}, i =f j.

Suppose that an optimal collection of all-l submatrices Rk x Ck C M

e

has been chosen,

U(Rk x Ck) =

e

M,

k=l

L (IRk I + ICkl) =

a(M). Let X

=

i=l

{XI, ... , xl'l, and let us define the following two sets for i = 1,2, ... , m:

Ai

.-

{xklriERk},

Bi

.-

{Xk

I Ci E Ck}.

Notice that the ordered pairs (i,j) with Ai n B j =f 0 correspond to precisely

those entries aij of M which occur in at least one Rk x Ck, therefore the

e

assumption

.

U(Rk x Ck) = M and the initial conditions on M imply that

k=l

(i)

(ii)

n Bi = 0

Ai n B j =f 0

Ai

for alII:::; i :::; m, and

or Aj

n Bi =f 0

for all i

=f j,

1:::; i,j :::; m.

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