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 [email protected]
Version: Tue Nov 11 08:48:41 2008
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Table of Contents
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A. Participant Contributions
1. 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
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Chapter A: Participant Contributions
A.1 Butkovic, Peter
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 λk → λ(A) where
λk is the Perron root of the Hadamard power Ak 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 b0 the system A ⊗ x = b0 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) = a0 ⊕ a1 ⊗ x ⊕ a2 ⊗ x(2) ⊕ ... ⊕ an ⊗ 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 ⊕ α1 ) ⊗ ... ⊗ (x ⊕ αn ) .
The values α1 , ..., αn 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.
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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 = {λ1 , λ2 , λ3 , λ4 }, for which λ1 > 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 λ1 so that we consider σ 0 = {1, r, a±ib},
how can we construct the set {x ∈ R3 : 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, 43–60. 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
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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 = (λ1 , ... , λn ) be a list of complex numbers and let sk := λk1 + ... + λkn ,
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 λj real and | λi |≤ λj , for all i. Such a λj is called a itPerron root
of σ.
A more obvious necessary condition is that all the sk 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 sk , 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 sk ≥ 0 for all positive integers k (and sm = 0for some m
implies sh = 0for all positive divisors h of m),then
σN := (λ1 ,
... , λn , 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 σN .
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 = (λ1 , ... , λn ) with Perron root λ1 is called
itPerron extreme if, for all ² > 0, (λ1 − ², λ2 , ... , λn ) 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 s1 = λ1 + ... + λn = 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.
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For example, if we call a realizable spectrum σ with Perron root λ1 (as above) very
extreme if s1 = 0, and, for all ² > 0, (λ1 − (n − 1)², λ2 + ², ... , λn + ²) 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) 249-316.
[2]J.E. Cohen and U.G. Rothblum. Nonnegative ranks, decompositionsand factorization of nonnegative matrices. LAA 190(1993) 149-168.
[3]T.J. Laffey. A sparsity result on nonnegative real matrices. Linear Operators Banach
Center Publ 38, pp 187-191. Polish Acad. Sc. 1997.
[4]T.J. Laffey. Extreme nonnegative matrices. LAA 275-276(1998) 349-357.
[5]T.J. Laffey and M.E. Meehan. A characterization of trace zerononnegative 5×5
matrices. LAA 302-303(1999) 295-302.
[6]T.J. Laffey and H. Šmigoc. Nonnegative realization of spectra havingnegative real
parts. LAA 416(2006) 148-159.
[7]T.J. Laffey and H. Šmigoc.A classical example in the nonnegativeinverse eigenvalue
problem. ELA 17(2008) 333-342.
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[8]R. Loewy and J. McDonald. The symmetric nonnegative inverseeigenvalue problem
for 5×5 matrices. LAA 393(2004) 275-298.
[9]M.E. Meehan. Some results on matrix spectra”, PhD thesis. NUI Dublin 1998.
[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) 729-773.
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 Rn . 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 Rn . 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.
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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 xi = T xi−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.
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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
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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, 2002.
(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, 1999. 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 (1908-1912) 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
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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.
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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 σ1 , . . . , σk , we would like to construct new realizable
lists. Below we present some specific questions on this topic.
A. Let (λ1 , a + ib, a − ib, λ3 , . . . , λn ) be the list of eigenvalues of some nonnegative matrix A, where λ1 denotes the Perron eigenvalue. Then (λ1 + 2t, a − t + ib, a − t −
ib, λ3 , . . . , λn ) is a list of eigenvalues of some nonnegative matrix for every t > 0,
[MR2223471,MR2344569].
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 (λ1 , λ2 , . . . , λn ) and let B
be a k × k principal submatrix of A with spectrum (µ1 , µ2 , . . . , µk ). Let the list
(µ1 , µ2 , . . . , µk , ν1 , ν2 , . . . , νs )
be realizable. Is then the list (λ1 , λ2 , . . . , λn , ν1 , ν2 , . . . , νs ) realizable? The same question can be posed for symmetric nonnegative matrices. Clearly both questions have
a positive answer when k = n. [MR2372590] 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
(λ1 , λ2 , . . . , λn , −µ1 , −µ2 , . . . , −µm )
7
is realizable.
C. Let (λ1 , λ2 , . . . , λk , λk+1 , . . . , λn ) be a realizable list of complex numbers with the Perron eigenvalue λ1 . For which lists of complex numbers (µ1 , µ2 , . . . , µs ) is the list
(µ1 , µ2 , . . . , µs , λk+1 , . . . , λn )
realizable?
When k = 1, we can take (µ1 , µ2 , . . . , µs ) 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 λ1 , [MR2098598].
Some results are known when k = 2, λ2 is real and λ1 > λ2 . Suppose that t1 and a
are nonnegative numbers and that t2 is a real number such that |t2 | ≤ t1 . If µ1 , µ2 , µ3
are roots of the polynomial
f (x) = (x − λ1 )(x − λ2 )(x − a) − x(t1 + t2 ) + t1 λ2 + t2 λ1 ,
then the list (µ1 , µ2 , . . . , µ3 , λ3 , . . . , λn ) is realizable, [MR2133312]. In particular, the
list
t(1 − t)λ22
(λ1 +
, tλ2 , (1 − t)λ2 , λ3 , . . . , λn )
λ1
is realizable for any t ∈ [0, 1], and the list
( λ22 )2 + β 2 λ2
λ2
,
+ iβ,
− iβ, λ3 , . . . , λn )
λ1
2
2
is realizable for any real number β.
(λ1 +
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 [MR1097240] 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.
[MR1350951] 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 [MR2290689] it is shown that (3 + t, 3 − t, 0, −2, −2, −2) is the spectrum of a
symmetric nonnegative matrix for t ≥ 31 . (It is not known if t = 13 is the smallest possible.)
Bibliography
[MR1097240] Boyle, M. and Handelman, D. (1991). The spectra of nonnegative matrices via
symbolic dynamics. Ann. of Math. (2), 133(2):249–316.
[MR2344569] Guo, S. and Guo, W. (2007). Perturbing non-real eigenvalues of non-negative
real matrices. Linear Algebra Appl., 426(1):199–203.
[MR1350951] 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):3647–3651.
8
[MR2223471] Laffey, T. J. (2004). Perturbing non-real eigenvalues of nonnegative real matrices. Electron. J. Linear Algebra, 12:73–76 (electronic).
[MR2290689] Laffey, T. J. and Šmigoc, H. (2007). Construction of nonnegative symmetric
matrices with given spectrum. Linear Algebra Appl., 421(1):97–109.
[MR2372590] Laffey, T. J. and Šmigoc, H. (2008). Spectra of principal submatrices of nonnegative matrices. Linear Algebra Appl., 428(1):230–238.
[Meehan] Meehan, E. (1998). Some Results On Matrix Spectra. PhD thesis, University
College Dublin.
[MR2098598] Šmigoc, H. (2004). The inverse eigenvalue problem for nonnegative matrices.
Linear Algebra Appl., 393:365–374.
[MR2133312] Šmigoc, H. (2005). Construction of nonnegative matrices and the inverse
eigenvalue problem. Linear Multilinear Algebra, 53(2):85–96.
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) 389–413. 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) 2435–2450) 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 Mmatrices 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.
9
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 PerronFrobenius 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 1954. A fundamental question is: How to characterize the set of
irreducible nonnegative matrices B such that there exists a nonnegative B̃ to satisfy:
B − B̃ = BB̃ = B̃B.
(0.1)
2) The Friedland-Karlin (FK) inequalities, a discrete analogue of the famous DonskerVaradhan 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,
Y
l
((Az)l /zl )xl yl ≥ ρ(A)
(0.2)
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 2008.
2) S. Friedland and C. W. Tan, Maximizing Sum Rates in Gaussian Interference-limited
Channels, arXiv, 0806(2860v2), 2008.
3) C. W. Tan, M. Chiang and R. Srikant, Fast Algorithms and Performance Bounds
for Sum Rate Maximization in Wireless Networks, submitted to IEEE Infocom 2009.
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 .
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, # 4:327-338, 2004
A Characterization of Positive Matrices, Charles Johnson and Pablo Tarazaga, Positivity Vol 9, #1:137-139, 2005.
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 Ak is a nonnegative matrix for all sufficiently large positive
integers k;
• eventually exponentially nonnegative if the matrix exponential etA 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 index0 (A) ≤ 1. What if index0 (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 eA 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).
11
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 ∆ = (λ1 , λ2 , ..., λn ) is the nonzero spectrum
of a strictly positive matrix if and only if
(a) λ1 > |λi | for all i > 2,
(b) tr∆k > 0 for all k = 1, 2, ...,
where tr∆k = λk1 + ... + λkn .
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.
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 AN +t ≥ 0 (componentwise) for some
N ≥ 0 and all t ≥ 0.
We denote by Jk (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 λ̄ be the complex conjugate of λ. A collection
U of elementary Jordan blocks is said to be self-conjugate if whenever Jk (λ) belongs to U
then so does the block Jk (λ̄), 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 ars 6= 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
·
¸
B C
T
P AP =
,
0 D
13
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 A12 0 · · ·
0
 0

0 A23 · · ·
0
 .

..
..
..
...
 ..
,
.
.
.


 0
0
0 · · · Am−1,m 
Am1 0
0 ···
0
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 Jk (λ) in U, the block Jk (e2πi/m λ)
(and hence also the blocks Jk (e2πri/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 e2πi/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(Ak ), 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 Am 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 ).
14
Comment 1. Given an irreducible nonnegative matrix P , then by the Frobenius
theorem and Wielandt’s lemma (see for instance [4, ch.XIII, Theorem 2 and lemma 2]), the
set of Jordan blocks U(P ) is a Frobenius collection. Therefore our assumption that U is
a Frobenius collection is natural. Frobenius collections impose no restrictions on nilpotent
Jordan blocks. In the proof of this theorem it is convenient to build a nonnegative irreducible
matrix starting from a Frobenius collection with no nilpotent blocks. A desired collection of
nilpotent blocks is added to the matrix in the final stage of the proof.
Comment 2. As follows from theorem BZ-BT an irreducible eventually nonnegative
matrix may have the null - space of any structure and any dimension. On the contrary, the
null spectrum of nonnegative matrices is subjected to some restrictions (see [3]). Condition
(d) suggests no restrictions on the null - space of irreducible nonnegative matrices as long
as only a part of nonsingular Jordan blocks and the spectral radius are defined a priori.
Comment 3. A Frobenius collection U may not satisfy the necessary properties of
nonnegative matrices [7]. For example, the condition tr(U k ) ≥ 0 may be violated. Therefore
some additional Jordan blocks must be added to U in order to get a Frobenius collection of
a nonnegative matrix. In other words, inclusions (b) and (c) are necessary and cannot be
replaced by equalities without additional assumptions.
Comment 4. The proof provides a constructive way to build nonnegative irreducible
matrices from Jordan blocks.
We will call nonnegative irreducible matrices with properties (a),(b), (c), (d) the nonnegative realizations of the self - conjugate Frobenius collections.
There is an open problem to find a nonnegative realization of minimal dimension, i.e.
the nonnegative minimal realizations.
Bibliography
[] A. Berman and R.J. Plemmons, Nonnegative Matrices in the Mathematical Sciences, SIAM
edition, SIAM, Philadelplia, 1994.
[2] A. Berman, M. Neumann and R.J. Stern, Nonegative Matrices in Dynamic Systems,
Wiley, New york, 1989.
[3] M. Boyle, D. Handelman, The spectra of nonnegative matrices via
symbolic dynamics, Annals of Math., 133 : 249–316 (1991).
[4] F.R. Gantmacher, The Theory of Matrices, Vol. 2, Chelsea, New York, 1959.
[5] E.B.Lee and L. Markus, Foundations of Optimal Control Theory, Wiley, New York, 1967.
[6] B.H. Lindquist, Asymptotic properties of powers of nonnegative matrices, with application, Linear Algebra and Applications 114/115 : 555-588 (1989).
[7] R. Loewy and D. London, A note on an inverse problem for nonnegative matrices, Linear
and Multilinear Algebra, 6: 83-90 (1978).
[8] H. Minc, Inverse elementary divisor problem for nonnegative matrices, Proc. Amer.
Math. Soc. 83 : 665–670 (1981).
[9] H. Minc, Inverse elementary divisor problem for doubly stochastic matrices, Linear and
Multilinear Algebra 11 : 121–131 (1982).
[10] H. Minc, Nonnegative Matrices, John Wiley & Sons, New York, 1988.
[11] Bo Henry Lindqvist, Asymptotic properties of powers of nonnegative matrices, with
Applications, Linear and Multilinear Algebra 114/115: 555 - 558 (1989).
15
[12] B.G. Zaslavsky, Eventually nonnegative realization of difference control systems. In:
Advanced Series Dynamical Systems, Vol. 9, Proceed. International Conference on Dynamical Systems and Related Topics Nagoya, 1990, edited by K. Shiraiwa, World Scientific, pp.
573-602,1991.
[13] B.G. Zaslavsky and Bit Shun Tam, On the Jordan form of an irreducible matrix with
eventually non-negative powers, Linnear Algebra and Its Applivation: 302/303: 303 - 330
(1999).