Nonnegative Matrix Theory: Generalizations and Applications

Transcription

1 Nonnegative Matrix Theory: Generalizations and Applications The American Institute of Mathematics The following compilation of participant contributions is only intended as a lead-in to the AIM workshop Nonnegative Matrix Theory: Generalizations and Applications. This material is not for public distribution. Corrections and new material are welcomed and can be sent to workshops@aimath.org Version: Tue Nov 11 08:48:

2 2 Table of Contents A. Participant Contributions Butkovic, Peter 2. Catral, Minerva 3. Eubanks, Sherod 4. Friedland, Shmuel 5. Gurvits, Leonid 6. Hall, Frank 7. Laffey, Thomas 8. Lins, Brian 9. Neumann, Michael 10. Olesky, Dale 11. Rothblum, Uriel 12. Schneider, Hans 13. Sergeev, Sergei 14. Smigoc, Helena 15. Soules, George 16. Szyld, Daniel 17. Tan, Chee Wei 18. Tarazaga, Pablo 19. Troitsky, Vladimir 20. Tsatsomeros, Michael 21. van den Driessche, Pauline 22. Zaslavsky, Boris

3 3 A.1 Butkovic, Peter Chapter A: Participant Contributions Let G =(G,, ) be a linearly ordered commutative group, a b = max(a, b) for a, b G and G = G {ε} where ε < a for all a G. Then (G,, ) is an idempotent commutative semiring. My research is in max-algebra, an analogue of linear algebra developed for the pair of operations (, ). If G = (R, +, ) [G = (R +,, )] then we speak about max-plus [max-times] algebra. An important characteristic of square matrices is the maximum cycle mean, denoted as λ(a). The eigenproblem A x = λ x and linear system problem A x = b are well known and efficiently solvable. In particular λ(a) is the unique eigenvalue of any irreducible matrix. 1. It is known that if A is an irreducible nonnegative matrix then k λ k λ(a) where λ k is the Perron root of the Hadamard power A k and λ(a) is in max-times. However A may have up to n independent max-algebraic eigenvectors and it is not immediately clear to which one the sequence of Perron eigenvectors is converging (if at all). We conjecture convergence to the barycentre of the set of fundamental max-algebraic eigenvectors. 2. I have recently shown that the following permutation problems in max-plus are N P -complete (when all entries are integer): (PEV) Given a square matrix A and a vector x, is it possible to permute the components of x so that the arising vector is an eigenvector of A? (PLS) Given a matrix A and a vector b, is it possible to permute the components of b so that for the arising vector b the system A x = b is solvable? We can of course formulate similar problems in conventional linear algebra. I have recently proved that both these problems are also N P -complete. However, PEV for positive matrices is easily solvable since by Perron-Frobenius there is a unique positive eigenvector (up to a multiple). This gives rise to the following open problems: (OP1) Is PEV for non-negative matrices polynomially solvable or N P -complete? (OP2) Is PLS for positive/non-negative matrices polynomially solvable or N P -complete? 3. An expression of the form p(x) = a 0 a 1 x a 2 x (2)... a n x (n) is called a maxpolynomial. It is known that each maxpolynomial considered as a function factorises to a product of n linear factors, that is p(x) = β (x α 1 )... (x α n ). The values α 1,..., α n are called corners of p(x). It is known that the greatest corner of the characteristic maxpolynomial for any square matrix A is λ(a). (OP3) What are the other corners? 4. I am also interested in the functions of matrices in max-algebra. A.2 Catral, Minerva I am interested in working on problems related to finite Markov chains and graphs. I have co-authored some papers in this direction. I have also done work on nonnegative matrix factorization, in particular symmetric nonnegative matrix factorization. I am also interested in problems related to generalizations of the Perron-Frobenius theory.

4 4 A.3 Eubanks, Sherod My interests and/or contributions focus on the nonnegative inverse eigenvalue problem (NIEP), with emphasis on the following: 1. Techniques for constructing realizing nonnegative matrices associated with both real and non-real spectra. 2. The nonnegative normal inverse eigenvalue problem (NNIEP). Specifically, we ask the question: Can every realiable list be realized by a nonnegative normal matrix, and if so, how to construct such a normal matrix? The answer is yes for matrices of order 3, but is open for order 4 and higher. 3. Which lists of 4 numbers σ = {λ 1, λ 2, λ 3, λ 4 }, for which λ 1 > 0 and the usual necessary conditions for NIEP on σ hold, are the spectrum of a nonnegative (normal or otherwise) matrix? If we divide each element of σ by λ 1 so that we consider σ = {1, r, a±ib}, how can we construct the set {x R 3 : x = (r, a, b)}? While some necessary and sufficient conditions exist for the 4 4 case, these conditions do not seem tractable enough to answer the latter question. 4. While some results regarding nonnegative normal matrices are known, little or nothing is known about the orthogonal matrices which yield the Schur form of these matrices. Some results are known, for example, in the case for which the orthogonal matrix is a Soules matrix. Some general constructive techniques for orthogonal matrices yielding normal or nonnegative matrices would be beneficial here, especially for the NNIEP. 5. Related to 4, are there any general properties of spectra of nonnegative normal matrices that are not satisfied by normal matrices that are not nonnegative? A.4 Friedland, Shmuel In 1978 I initiated a study of eventually nonnegative matrices in my paper On an inverse problem for nonnegative and eventually nonnegative matrices. Israel J. Math. 29 (1978), no. 1, I am still very much interested in many aspects of this inverse problem. Another interesting problem is an approximation of a given nonnegative m n matrix by a product of two nonnegative matrices XY, where X, Y are m k and k n respectively. A.5 Gurvits, Leonid I am interested in the stability of switched linear and nonlinear systems in as discrete as well continuous time. In particular, I am interested in the switched system with invariant pointed cones. Another topic, relevant to the workshop: a generalization of the permanent to the completely positive operators and the corresponding generalizations of Van der Waerden like conjectures. A.6 Hall, Frank I am generally interested in all 5 areas of the workshop. My particular interests are in spectral properties of nonnegative matrices, including characteristics associated with peripheral eigenvalues and eventually nonnegative matrices. There are a number of related papers in the literature. I hope that some of the lectures at the workshop will be focused on familiarizing the participants with the proper background, such as level characteristics and level forms. (It seems that definitions of some terms vary in some of the papers.) A development of the major known results would be good, as well as some illustrations. I hope

5 to do more reading and thinking before the workshop. I am appreciative of the efforts of the organizers and look forward to the workshop. A.7 Laffey, Thomas I. Nonnegative Inverse Eigenvalue Problem (NIEP) Let σ = (λ 1,..., λ n ) be a list of complex numbers and let s k := λ k λ k n, k = 1, 2, 3,... The NIEP asks for necessary and sufficient conditions on σ in order that σ be the spectrum of an entry-wise nonnegative matrix. If this occurs, If this occurs, we say that σ is itrealizable, and we call a nonnegative matrix A with spectrum σ a itrealizing matrix for σ. A necessary condition for realizability coming from the Perron-Frobenius theorem is that there exists j with λ j real and λ i λ j, for all i. Such a λ j is called a itperron root of σ. A more obvious necessary condition is that all the s k are nonnegative. In terms of n, a complete solution of the NIEP is only available forn 4. The solution for n = 4, expressed in terms of inequalities for the s k, appears in the PhD thesis of my former student ME Meehan[9] and a solution in terms of the coefficients of the characteristic polynomial has been published recently by Torre-Mayo et al.[10]. However, the same problem in which one may augment the list σ by appending an arbitrary number N of zeros was solved by Boyle and Handelman [1]. They proved the remarkable result that if σ has a Perron element and s k 0 for all positive integers k (and s m = 0for some m implies s h = 0for all positive divisors h of m),then 5 σ N := (λ 1,..., λ n, 0...., 0) (N zeros) is realizable. However, their proof is not constructive and does not provide a bound on the minimal number N = N(σ) required for realizability. One question I propose for investigation is Quention (1): Given σ satisfying the Boyle-Handelman conditions, find good upper and lower bounds on the minimum number N required for the realizability of σ N. Šmigoc and I [6] have obtained best possible results in the case that all elements of σ other than the Perron have non-positive real parts. More recently,in [7], we have obtained upper bounds onthe minimum N required in the case of the classic example σ = (3 + t, 3, 2, 2, 2). A concept that has proved useful in work on the NIEP is that of an extreme or (Perron extreme) spectrum. A realizable list σ = (λ 1,..., λ n ) with Perron root λ 1 is called itperron extreme if, for all ɛ > 0, (λ 1 ɛ, λ 2,..., λ n ) is not realizable. It follows from my work in [4], that if σis Perron extreme, then there exists a nonzero nonnegative matrix Y with AY = Y A and trace(ay ) = 0. If the trace s 1 = λ λ n = 0, then σ is obviously Perron extreme, but we can choose Y to be the identity matrix, and we get no useful information in this case. So we seek to find a more restrictive definition of extremality in the trace zero case. Question (2) Find a good concept of extremality and an associated investigative tool for trace zero spectra.

6 6 For example, if we call a realizable spectrum σ with Perron root λ 1 (as above) very extreme if s 1 = 0, and, for all ɛ > 0, (λ 1 (n 1)ɛ, λ 2 + ɛ,..., λ n + ɛ) is not realizable, then it is possible to get a somewhat analogous result,but further progress should be possible. Given a realizable spectrum σ, it is interesting to consider combinatorial properties of nonnegative matrices realizing it. For example, we find that it is possible to find realizing matrices with close to half their entries zero [3]. However, I expect that much stronger results are true. In the case of realizing spectra by nonnegative symmetric matrices, no sparsity result of this type is known in general, though for n 5, a similar result holds by work of Loewy and McDonald [8]. I propose the problem: Problem (3): Suppose that A is a nonnegative n n symmetric matrix. Find a bound for the least number of positive entries in symmetric nonnegative matrices cospectral with A. II. Nonnegative factorization of nonnegative matrices.. Suppose that A is a nonnegative n n matrix of rank r. The nonnegative factorization rank nfr(a) is the least positive integer k for which there exists a factorization A = BC where B is a nonnegative n k matrix and C is a nonnegativek n matrix. [See Cohen and Rothblum [2]] Clearly r nfr(a) n. Cohen and Rothblum have observed, inter alia, that if r = 2,then nfr(a) = 2. Beasley and I have observed that for r = 3, nfr(a) can be arbitrarily large. I propose the question Question (4): For every n 3, does there exist a nonnegative n n matrix A with rank(a) = 3 and nfr(a) = n. It would also be of interest to identify classes of nonnegative matrices A in which nfr(a) is bounded as a function of the rank r of A. Using a cone argument, Beasley and I have observed that this occurs if A is a product of two nonnegative matrices of rank r. Bibliography [1]M.Boyle and D. Handelman. The spectra of nonnegative matricesvia symbolic dynamics. Ann. Math.133(2)(1991) [2]J.E. Cohen and U.G. Rothblum. Nonnegative ranks, decompositionsand factorization of nonnegative matrices. LAA 190(1993) [3]T.J. Laffey. A sparsity result on nonnegative real matrices. Linear Operators Banach Center Publ 38, pp Polish Acad. Sc [4]T.J. Laffey. Extreme nonnegative matrices. LAA (1998) [5]T.J. Laffey and M.E. Meehan. A characterization of trace zerononnegative 5 5 matrices. LAA (1999) [6]T.J. Laffey and H. Šmigoc. Nonnegative realization of spectra havingnegative real parts. LAA 416(2006) [7]T.J. Laffey and H. Šmigoc.A classical example in the nonnegativeinverse eigenvalue problem. ELA 17(2008)

7 [8]R. Loewy and J. McDonald. The symmetric nonnegative inverseeigenvalue problem for 5 5 matrices. LAA 393(2004) [9]M.E. Meehan. Some results on matrix spectra, PhD thesis. NUI Dublin [10]J.Torre-Mayo, M.R. Abril-Raymundo,E. Alarcia-Estevez,C. Marijuan Lopez and M.Pisanero. The nonnegative inverse eigenvalue problem from the coefficients of the characteristic polynomial. EBL digraphs. LAA 426(2007) A.8 Lins, Brian I am interested in order-preserving homogeneous of degree one maps from a closed cone into itself. Of course, the classic example of such maps are the nonnegative matrices treated as maps from the closed cone of nonnegative vectors in R n. Many of the results from Perron-Frobenius theory can be extended from nonnegative matrices to nonlinear maps that are order-preserving and homogeneous of degree one. Some classes of nonlinear maps on cones are well understood, but many are not. I am interested in questions about the existence and uniqueness of eigenvectors in the interior of closed cones for several classes of nonlinear maps that are order-preserving and homogeneous of degree one. Maps that I am interested in include, reproduction-decimation operators defined on the cone of discrete Dirichlet forms, upper and lower transition operators associated with imprecise Markov chains, and certain averaging operators that are generalizations of the arithmetic-geometric mean operator. One can also ask about the asymptotic behavior of such order-preserving, homogeneous of degree one maps. Because the iterates of the map may be unbounded in norm, we typically focus on the normalized iterates. One open question of interest for several of these maps is to classify all possible orders of the periodic orbits. I am also interested in matrices whose inverses have strictly positive entries. Such matrices arise frequently in discrete nonlinear boundary value problems. The Birkhoff-Hopf theorem states that any matrix with all positive entries is a contraction in the Hilbert projective metric on the cone of nonnegative vectors in R n. The Birkhoff-Hopf theorem also gives a formula for calculating the contraction constant of a positive matrix. Unfortunately, the formula is computationally intensive. For many of the matrices that arise in discrete boundary value problems, the contraction constants for their inverses can be predicted using simple formulas. I would like to look at these problems and classify those inverse-positive matrices where simple expressions for the Birkhoff-Hopf contraction constant of the inverse can be given. A.9 Neumann, Michael Michael Neumann s research over the years can be divided into 5 main areas, each of which will be treated separately. (i) Iterative Methods for Exact and Approximate Solutions to Linear Systems of Equations Solving systems of linear equations is one of the best known applications of mathematics. It is employed in a host of problems of practical interest since the precise solutions to these problems are only implicitly known and the standard way to obtain a quantitative measurement of the solution is to approximate it at a finite number of input points. This approximation problem process usually results in a system of linear equations. 1

8 2 One of the techniques for solving such systems is an iteration scheme: generate a sequence of approximations, beginning with an initial guess, which will converge to the exact solution if an infinite number of steps were to be performed, or to a close approximate solution to the system if only a finite steps are performed. Neumann made an contribution to the subject by giving an exact characterization of all the major subspaces that are important in determining, for a system of linear equations Ax = b, and a fixed point of the iteration scheme x i = T x i 1 +c, the exact or approximate solution to which the iterates might converge. Prior to this work, knowledge of these subspaces was mostly confined to a narrow class of least squares problems. Neumann s work on the convergence of iterative methods for singular systems lead to his interest in the topic of nonnegative matrices because, in the presence of a nonnegative iteration matrix, much can be deduced about the necessary and the sufficient conditions for the convergence of an iteration scheme from both the numerical and the graph theoretical structure of the matrix. Neumann s papers on iterative methods deal also with conditions for convergence in the presence of iteration matrices which are not nonnegative, but have other recurring properties such as: paracontracting matrices, cyclic matrices, and positive definite matrices. Paracontracting matrices turned out to be particularly useful in parallel methods for solving systems of equations. (ii) Nonnegative Matrices. At the time Neumann became interested in applications of nonnegative matrices to iterative methods for solving linear systems Ax = b, R. S. Varga had produced a beautiful theory for their convergence in the case when A is nonsingular. The transition to singular systems necessitated understanding the deeper relation of a nonnegative matrix to its Perron eigenspace. It is here that Neumann made use of the existence of a nonnegative basis for the Perron eigenspace of a nonnegative matrix. One of the papers that Neumann wrote on this subject (with Robert J. Plemmons), has been cited many times in the literature. The proof of the existence of such a basis, due to Rothblum and Richman Schneider, had a combinatorial flavor to it, yet its use in iterative methods was in a noncombinatorial setting. After Neumann s use of the theorem of the existence of a nonnegative matrix in his work, Neumann wondered for 10 years whether there was an analytic (non combinatorial) proof of the existence of a nonnegative basis until it was found in 1988 in a joint work with Robert Hartwig and Nicholas Rose of NC State University. Neumann s work on nonnegative matrices also dealt with questions on convexity and concavity of the Perron root of a nonnegative matrix. The main tools used here were generalized inverses of singular and irreducible M matrices, on which he has written over 20 papers. One of the papers studies the form of the growth of the population in a population growth model due to British biologist P. H. Leslie. Quite surprising conclusions concerning the effect on the rate growth of the population due to change in fecundity at different ages in the population are found there. Yet another paper is an extension of Soules method: how can one construct a symmetric nonnegative matrices with a prescribed, but admissible, set of eigenvalues. This result is in the area of the inverse eigenvalue problem. Recently, Goldberger and Neumann were able to solve a conjecture which arose from the celebrated work on the inverse eigenvalue problem by Boyle and Handelman.

9 The work on the Soules bases has recently become useful in a new area of interest for Neumann, namely, learning the parts of objects, such as faces, through nonnegative matrix factorization. This is a method for reconstructing images of objects. Nonnegative matrix factorizations are different from other methods of reconstruction because of they make use of nonnegative constraints which allow additive approximations without cancellation. This means that parts of faces, such as nose, eyes, etc. emerge with a sharper focus when reconstructed. (iii) Nonnegative Dynamic Systems. In biological systems, such as predator prey problems, one can ask the question: what is the set of all initial states for an evolving population such that, beginning with each one of these states, the population will reach at some time in the future a state in which, from that time onwards, the size of no species will diminish. This problem is a special case of the so called cone reachability problem. It can be solved, by continuous time methods, as the solution to the corresponding dynamical system is a matrix exponential. Neumann and his colleague from Montreal, Ronald Stern, investigated this problem and developed an approach to its solution based on the eigenspace deflation process. Predator prey problems can often be cast as a system of linear differential equations. Suppose now that we try to solve the system of differential equations by means of a numerical approximation method, such as Euler s scheme. Then one question which has to be answered is this: If the continuous trajectory (i.e., solution) emanating from an initial state becomes at some time, and remains thereafter, nonnegative, then does a discrete trajectory, generated by, for example, Euler s scheme for estimating the solution to ordinary differential equations, also becomes nonnegative? The answer to this question involves some delicate issues, particularly about points that reach and stay in the boundary of the nonnegative quadrant. Using both asymptotic expansions and much eigenspace analysis, Neumann, with Stern and Tsatsomeros, showed that under certain mild restrictions on the size of the time steps, the continuous and the discrete reachability cones coincide. Eventually Neumann, with Stern and A. Berman, published a book on this topic entitled Nonnegative Matrices in Dynamic Systems. Over a 1000 copies of the book were sold and it has been quoted many times in the literature. (iv) Parallel Synchronous and Asynchronous Methods for Solving Linear Systems. In the mid 80 s papers began to appear in which parallel computing methods, mainly synchronous iteration scheme, for solving large linear systems of equations were suggested. In some of these methods the different operators which were successively applied were nonnegative matrices. At first, Neumann tried to build a framework for analyzing parallel methods. A paper on the subject, which he wrote with Plemmons, has been quoted over 100 times in the literature. Neumann then went on to investigate chaotic, i.e., asynchronous iteration methods. Here the first task was to try and express the procedure in some mathematical fashion. Once this was done, then a concept mentioned earlier, that of paracontracting operators, helped in proving convergence results. One of the more interesting papers (joint with Elsner and 3

10 4 Vemmer) proved that when increasing parallelization is applied to certain problems, then at first there is an improvement in the speed up rate of the computation, but beyond a certain point the rate levels off because of communication overheads. In recent years, Neumann has worked on parallel methods for computing, called divide and conquer schemes, objects of interest to practitioners working with probabilistic systems known as Markov chains. One such object of interest is the mean first time passage between various states of the system. Neumann delivered an invited talk on this subject at the conference on Computational Linear Algebra with Applications in Milovy, Czech Republic, August, (v) The Algebraic Connectivity of Graphs. There are types of matrices that can be used to model graphs. Then certain parameters associated with the matrices can tell us to what degree the graph is connected. One of these parameters involves the so called second smallest eigenvalue of the Laplacian matrix. This eigenvalue is known as the algebraic connectivity of a graph. However, the computation of this eigenvalue can be expensive and therefore it becomes necessary to approximate it well. In the last few years Neumann has been particularly engaged in developing sensitive, but cheap, estimators of this quantity. Such an approximation can be derived from the generalized inverse of the Laplacian matrix. The Laplacian matrix is, for connected graphs, another example where singular and irreducible M matrices appeared in Neumann s works. In 1995, at the Fourth International Linear Algebra Society in Atlanta, he gave a one hour invited talk tying applications of the generalized inverses of singular and irreducible matrices in iterative methods, Markov chains, eigenvalue perturbation problems, and graphs. He also gave an invited talk on this topic at the Haifa Eleventh Matrix Theory Conference, Haifa, Israel, June, Neumann has continued to work on these topics in the years since then. A.10 Olesky, Dale I have co-authored several recent papers concerning nonnegative matrices. One of the primary focuses of my research has been M- and inverse M-matrices, and, for example, positivity of principal minors. However, other recent papers have involved Perron-Frobenius theory, primitive matrices, totally positive matrices, permanents of (0,1)-matrices, and sign patterns that allow a positive or a nonnegative inverse. Some recent work has also involved eventually nonnegative matrices. A.11 Rothblum, Uriel The Perron Frobenius Theory of nonnegative matrices is a collection of results that refer to the spectrum and corresponding eigenvalues of nonnegative matrices. The earliest results of Perron (1908) and Frobenius ( ) refer to positive and irreducible matrices and assert that the spectral radius of such matrices is an eigenvalue having a strictly positive eigenvector which is a unique eigenvector, up to scalar multiple. There is a standard approach of extending part of the above results to nonnegative matrices. First, perturb the nonnegative matrix by converting each zero element to a fixed small positive element. Each resulting perturbed matrix is strictly positive and therefore has a strictly positive eigenvector corresponding to its spectral radius; of course, such eigenvectors

11 can be normalized to have norm 1. Next, reduce the perturbation parameter to zero; a convergence argument then shows that a limit point of the normalized eigenvectors is a nonzero nonnegative eigenvector of the unperturbed matrix that corresponds to its spectral radius. When restricting this conclusion to positive matrices, it produces weaker results than those available (and mentioned above) for such matrices. First, semi-positivity is weaker than strict positivity. Second, there is not reference to uniqueness of the eigenvector. Thus, the restriction of results obtained by the above convergence argument to nonnegative matrices which happen to be positive produces weaker results than those that are available from direct analysis of positive matrices. This gap was closed in work of Rothblum (1975) and Richman and Schneider (1978) through the examination of the generalized eigenspace of nonnegative matrices corresponding to their spectral radius. It was shown how the examination of the class structure of nonnegative matrices can produce a basis of nonnegative generalized eigenvectors with complete information about zero and nonzero elements. Generically, such basis are nowadays called preferred basis of nonnegative matrices. Typical proofs of these results do not use a perturbation argument. Single-parameter-perturbations of nonnegative matrices are known to generate a lot of interesting characteristics of matrices. In particular Laurent and Puseux expansions of the resolvent, eigenvalues and eigenvectors are then available. Of course, letting the perturbation parameter converge to zero, eliminates much of the information that is present in the perturbed characteristics. Examining special instances of nonnegative matrices suggests that the information about the structure of the eigenspace of nonnegative matrices and their preferred basis can be generated from the characteristics (spectral radius and eigenvector) of positive perturbations of those matrices. A.12 Schneider, Hans Nonnegative matrix theory can be investigated from several points of view: analytic, algebraic, geometric and graph theoretic. The emphasis may be theoretical or algorithmic. It s my desire to reinforce communication among researchers who have considered the subject from different points of view. In particular, I am interested in bringing together researches who have studied Perron-Frobenius theory in classical and max linear algebra. Until recently, the latter was developed almost independently of the former, yet many results have analogues. Both classical and max linear operators may also be considered in a more general context, namely that of homogeneous, monotonic operators on a finite dimensional real space. A.13 Sergeev, Sergei My main interests are in the area of max algebra and tropical convexity. This means investigating algebraic and convex-like properties of such subsets of the nonnegative orthant of the finite dimensional real space that are closed under componentwise maximisation and scalar multiplication. In max algebra, one extends the arithmetic operations := max and = to matrices and vectors, in order to examine the behavior of sequences A k x, or to solve max-linear systems A x = B x and A x = B y, or to find eigenvalues and eigenvectors of matrices and less trivial operators like cyclic projectors and general min-max functions. In tropical convexity, my main interests have been in the max analogues of the known facts from the convex geometry, and also in relations between tropical and ordinary convexity. 5

12 6 The following problems are of interest to me: 1. The generalised eigenproblem A x = λb x over max-algebra: it lacks both general theory and good algoritms for computing both eigenvectors and eigenvalues. I am also interested in the analogous problem over nonnegative matrix algebra, what techniques have been developed and how they can apply in the case of max algebra. 2. Eigenvalues and eigenvectors of cyclic projectors. These specific homogeneous nonlinear operators, recently investigated in my work S. Sergeev Multiorder, Kleene stars and cyclic projectors in max algebra (posted on arxiv.org) seem to be important in max algebra since they lead to approximations, in the sense of the Hilbert projective metric, of solutions to two-sided systems A x = B x and A x = B y, and multisided systems A x = B y = C z =... The eigenvalues have been described, and there are some algorithms which solve spectral problems for such operators. However, the theory of eigenvectors is yet not sufficient. 3. Max algebra is related to nonnegative matrix scaling problems. In particular, the scaling which visualises maximum cycle geometric mean of a nonnegative matrix is an important tool in analysing the behaviour of max-algebraic powers of matrices, due to the works of L. Elsner and P. van den Driessche. Another important nonnegative matrix scaling which seems to be strongly related to max algebraic problems but has not been investigated from this point of view, is the max balancing which appeared in the works of H. Schneider. A.14 Smigoc, Helena We will present some problems connected with the nonnegative inverse eigenvalue problem, the problem of finding necessary and sufficient conditions on a list of complex numbers σ in order that it be the spectrum of a nonnegative n n matrix. We will call a list of complex numbers σ realizable, if there exists a nonnegative matrix with spectrum σ. Operations that preserve realizability. What operations on a realizable list σ preserve its realizability? More generally, using given realizable lists of complex numbers σ 1,..., σ k, we would like to construct new realizable lists. Below we present some specific questions on this topic. A. Let (λ 1, a + ib, a ib, λ 3,..., λ n ) be the list of eigenvalues of some nonnegative matrix A, where λ 1 denotes the Perron eigenvalue. Then (λ 1 + 2t, a t + ib, a t ib, λ 3,..., λ n ) is a list of eigenvalues of some nonnegative matrix for every t > 0, [MR ,MR ]. Can one find operations involving imaginary parts of eigenvalues that preserve realizability of the list? B. Let A be an irreducilbe nonnegative matrix with spectrum (λ 1, λ 2,..., λ n ) and let B be a k k principal submatrix of A with spectrum (µ 1, µ 2,..., µ k ). Let the list (µ 1, µ 2,..., µ k, ν 1, ν 2,..., ν s ) be realizable. Is then the list (λ 1, λ 2,..., λ n, ν 1, ν 2,..., ν s ) realizable? The same question can be posed for symmetric nonnegative matrices. Clearly both questions have a positive answer when k = n. [MR ] show that the questions have a positive answer when k = 1, and in some other special cases. In particular, it is shown that the list (λ 1, λ 2,..., λ n, µ 1, µ 2,..., µ m )

13 is realizable. C. Let (λ 1, λ 2,..., λ k, λ k+1,..., λ n ) be a realizable list of complex numbers with the Perron eigenvalue λ 1. For which lists of complex numbers (µ 1, µ 2,..., µ s ) is the list (µ 1, µ 2,..., µ s, λ k+1,..., λ n ) realizable? When k = 1, we can take (µ 1, µ 2,..., µ s ) to be any realizable list of complex numbers that can be realized by a nonnegative matrix with a diagonal element that is greater than or equal to λ 1, [MR ]. Some results are known when k = 2, λ 2 is real and λ 1 > λ 2. Suppose that t 1 and a are nonnegative numbers and that t 2 is a real number such that t 2 t 1. If µ 1, µ 2, µ 3 are roots of the polynomial f(x) = (x λ 1 )(x λ 2 )(x a) x(t 1 + t 2 ) + t 1 λ 2 + t 2 λ 1, then the list (µ 1, µ 2,..., µ 3, λ 3,..., λ n ) is realizable, [MR ]. In particular, the list (λ 1 + t(1 t)λ2 2 λ 1, tλ 2, (1 t)λ 2, λ 3,..., λ n ) is realizable for any t [0, 1], and the list (λ 1 + ( λ 2 2 ) 2 + β 2 λ 1, λ iβ, λ 2 2 iβ, λ 3,..., λ n ) is realizable for any real number β. Effect of adding zeros to the spectrum in the symmetric case. Given a list of n complex numbers σ, in a remarkable piece of work [MR ] found necessary and sufficient conditions in order that σ with sufficiently many zeros adjoined be realizable as the spectrum of a nonnegative matrix. Their proof, via ergodic theory and the theory of shifts of finite type, is not constructive and does not give information on the minimum number of zeros which need to be appended to make the list realizable. In the symmetric case the effect of zeros added to the spectrum is not well understood. [MR ] shows that if this form of realizability cannot be achieved by the appending of n(n + 1)/2 zeros, then appending further zeros does not help. The smallest t for which (3 + t, 3 t, 2, 2, 2) is the spectrum of a symmetric nonnegative matrix is t = 1. This was shown by Loewy and the proof can be found in [Meehan]. In [MR ] it is shown that (3 + t, 3 t, 0, 2, 2, 2) is the spectrum of a symmetric nonnegative matrix for t 1. (It is not known if t = 1 is the smallest possible.) 3 3 Bibliography [MR ] Boyle, M. and Handelman, D. (1991). The spectra of nonnegative matrices via symbolic dynamics. Ann. of Math. (2), 133(2): [MR ] Guo, S. and Guo, W. (2007). Perturbing non-real eigenvalues of non-negative real matrices. Linear Algebra Appl., 426(1): [MR ] Johnson, C. R., Laffey, T. J., and Loewy, R. (1996). The real and the symmetric nonnegative inverse eigenvalue problems are different. Proc. Amer. Math. Soc., 124(12):

14 8 [MR ] Laffey, T. J. (2004). Perturbing non-real eigenvalues of nonnegative real matrices. Electron. J. Linear Algebra, 12:73 76 (electronic). [MR ] Laffey, T. J. and Šmigoc, H. (2007). Construction of nonnegative symmetric matrices with given spectrum. Linear Algebra Appl., 421(1): [MR ] Laffey, T. J. and Šmigoc, H. (2008). Spectra of principal submatrices of nonnegative matrices. Linear Algebra Appl., 428(1): [Meehan] Meehan, E. (1998). Some Results On Matrix Spectra. PhD thesis, University College Dublin. [MR ] Šmigoc, H. (2004). The inverse eigenvalue problem for nonnegative matrices. Linear Algebra Appl., 393: [MR ] Šmigoc, H. (2005). Construction of nonnegative matrices and the inverse eigenvalue problem. Linear Multilinear Algebra, 53(2): A.15 Soules, George Inequalities involving the permanent of either nonnegative or positive semidefinite matrices, including bounds on the permanent of nonnegative matrices, and the permanent-ontop conjecture (a special case of the Soules conjecture) for psds matrices. Pate s recemt results on this topic. The inverse eigenvalue problem for nonnegative symmetric matrices; Soules matrices. Hidden Markov models and the P-Q inequality. Duality for concave programs. A.16 Szyld, Daniel Joint submission of Abed Elhashash and Daniel B. Szyld. Matrices with Perron-Frobenius properties, which are not nonnegative We have studied matrices with a Perron-Frobenius property, such as having the spectral radius as an eigenvalue, and the corresponding eigenvector being nonnegative. We have characterized sets of matrices with Perron-Frobenius properties in the paper On general matrices having the Perron-Frobenius property, Electronic Journal on Linear Algebra, vol. 17 (2008) For example, the set WPFn is the set n n real matrices whose spectral radius is a positive eigenvalue having nonnegative left and right eigenvectors. In a subsequent paper ( Generalizations of M-matrices which may not have a nonnegative inverse, Linear Algebra and its Applications, vol. 249 (2008) ) we study several generalizations of M-matrices. For example, a GM-matrix A is of the for A = si B, where B is in WPFn, and ρ(b) s. We have shown that in some cases, well-known results of nonnegative matrices and M- matrices carry over to these matrices. In other cases, analogous results can be obtained. In particular, some necessary conditions for the existence of an inverse M-matrix were derived, using GM-matrices. We are interested in continuing the study of these matrices. For example, we are interested to know something about the functions which leave invariant these new sets of matrices. Other problems of interest deal with possible comparison theorems for splittings of GM-matrices and similar generalizations of M-matrices.

15 A.17 Tan, Chee Wei Application of Nonnegative Matrix Theory to Nonconvex Optimization Nonnegative matrix theory has a wide range of applications. We study the intriguing relationship between irreducible nonnegative matrix theory and nonconvex optimization that are motivated by communication theory, network optimization theory and engineering applications. Our recent work show how powerful tools in nonnegative matrix theory can be applied to these nonconvex problems by transforming them into eigenvalue optimization problems and give engineering insights. The principal nonnegative matrix theory tools that we use in our work are the Perron- Frobenius Theorem, Kingman s log-convexity, Friedland-Karlin inequalities and Wong s quasiinvertibility. We describe briefly below two such tools and their open questions. 1) The quasi-invertibility notion of nonnegative matrices was proposed by Wong for mathematical economics in A fundamental question is: How to characterize the set of irreducible nonnegative matrices B such that there exists a nonnegative B to satisfy: B B = B B = BB. (0.1) 2) The Friedland-Karlin (FK) inequalities, a discrete analogue of the famous Donsker- Varadhan variational formula that was derived in 1975 and later extended in 2008, proves to be extremely useful in tackling nonconvex network optimization and communication theory problems. One such FK inequality is given by: For any irreducible nonnegative matrix A, ((Az) l /z l ) x ly l ρ(a) (0.2) l for all strictly positive z, where x and y are the Perron and left eigenvectors of A respectively. Equality holds if and only if z = ax for some positive a. An interesting question is: are there tighter bounds when z can have zero entries (but not knowing which entry a priori)? The applications of the above tools illustrate new theoretical methodologies (exploit the eigenspace of quasi-inverse matrix, use FK inequalities to bound nonconvexity) and computational frameworks to solve certain class of NP-hard problems. Future research can take many directions, including examine the role and consequences of cone nonnegativity to nonconvex optimization. References of our work include: 1) C. W. Tan, Nonconvex Power Control in Multiuser Communication Systems, Ph.D. Dissertation, Princeton University, Princeton, N.J., USA, November ) S. Friedland and C. W. Tan, Maximizing Sum Rates in Gaussian Interference-limited Channels, arxiv, 0806(2860v2), ) C. W. Tan, M. Chiang and R. Srikant, Fast Algorithms and Performance Bounds for Sum Rate Maximization in Wireless Networks, submitted to IEEE Infocom A.18 Tarazaga, Pablo Matrices with the Perron-Frobenius property are defined in different ways. We will say that a matrix A has the Perron-Frobenius property if ρ(a) is an eigenvalue and the associated eigenvector/s is/are nonnegative. We will denote this set by P F. 9

16 10 It is now well known that P F is larger than the nonnegative orthant. Some sufficient conditions were given for a more restrictive Perron-Frobenius property (ρ(a) is a simple eigenvalue and the eigenvectors associated are positive) during last years. We are now interested in a description of P F and in new sufficient conditions for this property that may arise from the geometry of the set. References: Perron-Frobenius Theorem for Matrices with some Negative Entries, Pablo Tarazaga, Marcos Raydan and Ana Hurman, Linear Algebra and its Applications, 328:57-68, (2001) On Matrices with Perron-Frobenius Propertyand some Negative Entries, Charles Johnson and Pablo Tarazaga, Positivity Vol 8, # 4: , 2004 A Characterization of Positive Matrices, Charles Johnson and Pablo Tarazaga, Positivity Vol 9, #1: , A.19 Troitsky, Vladimir I am interested in extensions of Perron-Frobenius theory to operators and semigroups of operators on Banach lattices. A.20 Tsatsomeros, Michael Recall that an n n matrix A is called: eventually nonnegative if A k is a nonnegative matrix for all sufficiently large positive integers k; eventually exponentially nonnegative if the matrix exponential e ta is a nonnegative matrix for all sufficiently large t > 0. I think I can best describe my specific interests in this workshop with some direct questions: Under what conditions does eventual nonnegativity imply eventual exponential nonnegativity? It does when index 0 (A) 1. What if index 0 (A) 2? Under what conditions does eventual exponential nonnegativity imply eventual nonnegativity? How are the notions of eventual exponential nonnegativity of A and eventual nonnegativity of e A related? Let A be an eventually nonnegative matrix. When does there exist a > 0 such that A + ai is eventually nonnegative? When is A + ai eventually nonnegative for all a > 0? How does one detect an eventually (exponentially) nonnegative matrix? What can the combinatorial structure of an eventually nonnegative matrix be? By combinatorial structure here I mean the (signed) directed graph and the reduced graph. What are the sign patterns that allow or require eventual nonnegativity? A.21 van den Driessche, Pauline Several of my research interests are related to Nonnegative Matrix Theory: Generalizations and Applications. These I list below (in no particular order).

17 M-matrices and their inverses, especially as they arise in applications to models in mathematical biology. Exponents of primitive matrices as represented by 0,1 matrices and digraphs. Sign pattern problems, for example, patterns that allow a positive or nonnegative left inverse. Problems in the max algebra, especially those that either use results from the classical nonnegative matrix theory or lead to results in the classical algebra. I would especially like to learn about and work on recent new applications of nonegative matrix theory, in particular those used in biological and social sciences. A.22 Zaslavsky, Boris Nonnegative irreducible matrices with given subset of Jordan blocks and nonnegative realization problem Abstract. The Perron - Frobenius theory describes the properties of nonsingular Jordan blocks of cyclic irreducible nonnegative matrices. We call a set of Jordan blocks with such properties a self - conjugate Frobenius collection. Given a self - conjugate Frobenius collection of nonsingular Jordan blocks with cyclic index m and spectral radius ρ, we construct a cyclic irreducible nonnegative matrix with cyclic index m and spectral radius ρ that includes in the set of its Jordan blocks all Jordan blocks of this collection and has a given set of nilpotent Jordan blocks. AMS classification 15A21; 15A42, 93B15; 93B60 Keywords: Cyclic index; Irreducible eventually nonnegative matrix; cyclic matrix; nonnegative realization; Self - conjugate Frobenius collection; Solid convex cone 1. Introduction Many efforts have been carried out to resolve the famous Nonnegative Inverse Eigenvalue Problem [1,3,7 and further]. The detail discussion can be found in [13]. The strongest achievement in the field is the M. Boyle and D. Handelman s theorem. We will present it below in the simplified form for the ring of complex numbers: Theorem. The set of complex numbers = (λ 1, λ 2,..., λ n ) is the nonzero spectrum of a strictly positive matrix if and only if (a) λ 1 > λ i for all i > 2, (b) tr k > 0 for all k = 1, 2,..., where tr k = λ k λ k n. Although this is a very powerful result, some very important questions remain to be open: 1. What are the properties of the null -space of cyclic irreducible nonnegative matrices? 2. What are the properties of nonsingular Jordan blocks of cyclic irreducible nonnegative matrices? If replace the adjective nonnegative with the words eventually nonnegative, and thus consider a broader class of matrices, B. Zaslavsky - Bit-Shun Tam theorem 5.1 [13] answers to questions 1-3. The current paper is an extension of results [13] on cyclic irreducible nonnegative matrices and therefore it belongs to the Nonnegative Inverse Elementary Divisors Problem [8-10]. The paper is organized as follows. 11

18 12 In Section 2 we introduce the necessary definitions and notation. In particular, we present the notion of a Frobenius collection of elementary Jordan blocks. We remind the one - to - one correspondence between Frobenius collections of Jordan blocks and eventually nonnegative matrices. In Section 3 we associate with a given m - cyclic Frobenius collection of elementary Jordan blocks a m - cyclic irreducible nonnegative matrix. The spectral radius of the Frobenius collection is the spectral radius of the nonnegative matrix. For each nonsingular Jordan block of the given Frobenius collection there is an identical Jordan block of the nonnegative matrix. The set of nilpotent Jordan blocks of the Frobenius collection equals to the set of nilpotent blocks of the nonnegative matrix. In Section 4 we compare the result with the Boyle - Handelman theory and illustrate the result with numerical examples. In Section 5 we provide the control theory application of the result. In particular, we match the eventually nonnegative control systems with the nonnegative control systems. The trajectories of eventually nonnegative control systems are the projections of the trajectories of nonnegative control systems. 2. Irreducible eventually nonnegative matrices In order to describe our main result, we need to borrow the definitions and notations from paper [13]. A matrix A is called eventually nonnegative if A N+t 0 (componentwise) for some N 0 and all t 0. We denote by J k (λ) the k k upper triangular elementary Jordan block associated with the eigenvalue λ. For any complex square matrix A, the spectrum, the spectral radius, and the collection of elementary Jordan blocks associated with A are denoted respectively by σ(a), ρ(a) and U(A). Here we treat σ(a) and U(A) as multi-sets, the repetition number of an element of σ(a) being its algebraic multiplicity as an eigenvalue of A, and the repetition number of an elementary Jordan block in U(A) being the number of times the block occurs in the Jordan form of A. By the peripheral spectrum of A we mean the set which consists of eigenvalues of A with modulus ρ(a). Given a (finite nonempty) collection U of (not necessarily distinct) elementary Jordan blocks, by the spectrum (radius) of U, denoted by σ(u) (ρ(u)) we mean σ(a) (ρ(a)), where A is any matrix for which U(A) = U. Let λ be a non-real complex number and λ be the complex conjugate of λ. A collection U of elementary Jordan blocks is said to be self-conjugate if whenever J k (λ) belongs to U then so does the block J k ( λ), and the two blocks occur the same number of times in U. For a real square matrix A clearly U(A) is a self-conjugate collection of elementary Jordan blocks. For an n n matrix A, by the digraph of A, denoted by G(A), we mean the directed graph with vertex set {1,, n} such that (r, s) is an arc if and only if a rs 0, (r, s = 1,, n). We call an n n complex square matrix A irreducible if its digraph G(A) is strongly connected; or equivalently, if n = 1, or n 2 and there does not exist a permutation matrix P such that P T AP = [ B C 0 D ],

19 where B, D are nonempty square matrices. We call a square matrix m-cyclic if it is permutationally similar to a matrix of the form 0 A A A, m 1,m A m where the zero blocks along the main diagonal are square. The largest positive integer m for which A is m-cyclic is called the cyclic index of A. A collection U of elementary Jordan blocks with ρ(u) > 0 is said to be m-cyclic provided that for any nonsingular elementary Jordan block J k (λ) in U, the block J k (e 2πi/m λ) (and hence also the blocks J k (e 2πri/m λ) for r = 2,, m 1) belongs to U, and the two blocks occur the same number of times in U. [Equivalently, U is m-cyclic if ρ(u) > 0 and for any (or, for some) A with U(A) = U, A is similar to e 2πi/m A [13].] The largest m for which U is m-cyclic is called the cyclic index of U. We call a collection U of elementary Jordan blocks Frobenius [13] if for some positive integer m the following set of conditions is satisfied : (a) ρ(u) > 0, and there is exactly one block in U associated with ρ(u) and this block is 1 1. (b) If λ σ(u) and λ = ρ(u), then λ must be ρ(u) times an m th root of unity. (c) U is m-cyclic. Note that the definition of a Frobenius collection (and also of the cyclic index of a collection) of elementary Jordan blocks does not depend on its nilpotent members. For a collection U of elementary Jordan blocks, and for any positive integer k, we denote by U k the collection U(A k ), where A is any square complex matrix that satisfies U(A) = U and by tr(u) = tra. Theorem BZ-BT [13]. Let U be a collection of elementary Jordan blocks. The following conditions are equivalent : (a) U is a self-conjugate Frobenius collection with cyclic index m. (b) There exists an m-cyclic irreducible eventually nonnegative matrix A such that U(A) = U, and A m is permutationally similar to a direct sum of m eventually positive matrices. 3. Irreducible nonnegative matrices Now we can formulate the main result. Theorem 3.1. Given a self-conjugate Frobenius collection U of nonsingular Jordan blocks. Let m be the cyclic index and ρ be the spectral radius of the collection. Given an arbitrary set Z of nilpotent Jordan blocks. Then there exists a m-cyclic irreducible nonnegative matrix P such that: (a) ρ = ρ(p ), (b) σ(u) σ(p ), (c) U U(P ), (d) Z is the subset of all nilpotent blocks of U(P ). 13