maximal exponents of polyhedral cones iii article history received 31 may 2009 available online 14 november 2009 submitted by j bastero

unicity of minimal rank completions for tri diagonal partial block matrices abstract in this paper we establish necessary and sufficient conditions for a minimal rank completion of a tri diagonal partial block matrix to be unique the case of 4 4 block matrices with four consecutive diagonals is also considered along with its application to the problem of completing a matrix and its inverse

minimal representations of semiseparable kernels and systems with separable boundary conditions abstract the simplest representations of triangular parts of finite rank kernels are analysed criteria for uniqueness up to similarity are given the results are applied to the problem of minimal realization of systems with separable boundary conditions

unique minimal rank extensions of triangular operators the problem to find the minimal rank extensions of the lower triangular part of a finite rank matrix kernel appears in a natural way see 3 in the study of minimal representations of semi separable integral operators or equivalently of minimal realizations of boundary value systems in particular one is interested in those kernels for which the minimal rank extension is unique in this paper we give a full description of this class of so called unique matrix kernels we also define and characterize lower uniqueness for finite and semi infinite block operator matrices to state one of our main results let k t s be an m x n matrix kernel defined on the square a b x a b following 3 we say that k is lower separable if the lower triangular part k of k which is defined by kl t 8 k t s a s t b admits a finite rank extension i e there exists a finite rank matrix kernel h on a b x a b such that k is the lower triangular part of h recall that for an m x n matrix kernel h on a fl x y s the rank of

on the distinguished eigenvalues of a cone preserving map abstract we generalize many known results on a nonnegative matrix concerning linear inequalities collatz wielandt sets and generalized eigenvectors to the setting of a matrix preserving a finite dimensional proper cone a simple cone theoretic proof is given for the nonnegative basis theorem for the algebraic eigenspace of a nonnegative matrix the result is also extended to a matrix preserving a polyhedral cone given proper cones k 1 and k 2 in different euclidean spaces a necessary and sufficient condition is also obtained for the existence of a nonzero matrix x which takes k 2 into k 1 and satisfies ax xb where a b are given matrices preserving k 1 and k 2 respectively this extends and answers a recent open question posed by hartwig

toeplitz minimal rank completions it is proven that for banded patterns a block toeplitz partial matrix has a minimal rank completion which is block toeplitz the results are applied to a variation of the partial realization problem

two cores of a nonnegative matrix we prove that the sequence of eigencones i e cones of nonnegative eigenvectors of positive powers a k of a nonnegative square matrix a is periodic both in max algebra and in nonnegative linear algebra using an argument of pullman we also show that the minkowski sum of the eigencones of powers of a is equal to the core of a defined as the intersection of nonnegative column spans of matrix powers also in max algebra based on this we describe the set of extremal rays of the core the spectral theory of matrix powers and the theory of matrix core is developed in max algebra and in nonnegative linear algebra simultaneously wherever possible in order to unify and compare both versions of the same theory

the structure of the algebraic eigenspace to the spectral radius of eventually compact nonnegative integral operators let k be an eventually compact linear integral operator on lp 1 p with nonnegative kernel k x y where the underlying measure is totally finite on the domain set when p 1 this work extends the previous analysis of the author who characterized the distinguished eigenvalues of k and k and the support sets for the eigenfunctions and generalized eigenfunctions belonging to the spectral radius of k or k the characterizations of the support sets for the algebraic eigenspaces of k or k are phrased in terms of significant k components which are maximal irreducible subsets of and which yield a positive spectral radius for the integral operator defined by the restriction of k x y to the cartesian product of such sets in this paper we show that a basis for the functions constituting the algebraic eigenspaces of k and k belonging to the spectral radius of k can be chosen to consist of elements which are positive on their sets of support except possibly on sets of measure less than some arbitrarily specified positive number in addition we present necessary and sufficient conditions in terms of the significant k components for both k and k to possess a positive eigenfunction a e corresponding to the spectral radius as well as necessary and sufficient conditions for the sequence nkng p to converge whenever g 0 where p denotes the norm in lp and 1 the smallest in modulus characteristic value of k this analysis is made possible by introducing the concepts of chains lengths of chains height and depth of a significant k component as was done by u rothblum lin alg appl 12 1975 281 292 for the matrix setting

more on maximal and minimal ranks of schur complements with applications the maximal and minimal ranks of the schur complement d cgb are determined with respect to generalized inverse g of a matrix a where g is taken respectively as least squares g inverse a 1 3 minimum norm g inverse a 1 4 as well as a 1 2 3 a 1 2 4 and a 1 3 4 various consequences are also presented

minimal rank completions for block matrices abstract this paper deals with the following problem of completion of block matrices let aij i j be given matrices find additional matrices aij such that the completion a aij ni j 1 has lowest possible rank the structure of the set of all minimal rank completions is studied special attention is paid to the toeplitz case together with its connection to the partial realization problem

a study of projectionally exposed cones abstract in view of possible applications to abstract convex programs barker laidacker and poole have made an initial study of p exposed cones and o p exposed cones by restricting their attention to closed pointed cones a study of these cones is continued in this paper sufficient conditions for a face of a proper cone to be a p exposed or an o p exposed face are established characterizations of o p exposed cones among polyhedral cones are also obtained the relation between exposedness and p exposedness of faces of a general cone is examined as one consequence a question posed by iochum is answered in the finite dimensional case that is there is no finite dimensional semiregular self dual cones which are not regular also a conjecture posed by barker and thompson on p exposed faces of p n that the cone of all real polynomials of degree n which are nonnegative on the closed interval 0 1 is settled

theory of cones abstract this survey deals with the aspects of archimedian partially ordered finite dimensional real vector spaces and order preserving linear maps which do not involve spectral theory the first section sketches some of the background of entrywise nonnegative matrices and of systems of inequalities which motivate much of the current investigations the study of inequalities resulted in the definition of a polyhedral cone k and its face lattice f k in section ii a the face lattice of a not necessarily polyhedral cone k in a vector space v is investigated in particular the interplay between the lattice properties of f k and geometric properties of k is emphasized section ii b turns to the cones k in the space of linear maps on v recall that k is the cone of all order preserving linear maps of particular interest are the algebraic structure of k as a semiring and the nature of the group aut k of nonsingular elements a k for which a 1 k as well in a short final section the cone p n of n n positive semidefinite matrices is discussed a characterization of the set of completely positive linear maps is stated the proofs will appear in a forthcoming paper

completing a block diagonal matrix with a partially prescribed inverse abstract a complete solution of the matrix completion problem a b 1 c d is obtained in terms of solutions of a riccati type matrix equation several special cases are described in detail

on completions of symmetric and antisymmetric block diagonal partial matrices abstract a partial matrix is a matrix where only some of the entries are given we determine the maximum rank of the symmetric completions of a symmetric partial matrix where only the diagonal blocks are given and the minimum rank and the maximum rank of the antisymmetric completions of an antisymmetric partial matrix where only the diagonal blocks are given

on the invariant faces associated with a cone preserving map for an n n nonnegative matrix p an isomorphism is obtained between the lattice of initial subsets of 1 n for p and the lattice of p invariant faces of the nonnegative orthant r motivated by this isomorphism we generalize some of the known combinatorial spectral results on a nonnegative matrix that are given in terms of its classes to results for a cone preserving map on a polyhedral cone formulated in terms of its invariant faces in particular we obtain the following extension of the famous rothblum index theorem for a nonnegative matrix if a leaves invariant a polyhedral cone k then for each distinguished eigenvalue ofa fork there is a chain of m distinct a invariant join irreducible faces of k each containing in its relative interior a generalized eigenvector of a corresponding to referred to as semi distinguished a invariant faces associated with where m is the maximal order of distinguished generalized eigenvectors of a corresponding to but there is no such chain with more than m members we introduce the important new concepts of semi distinguished a invariant faces and of spectral pairs of faces associated with a cone preserving map and obtain several properties of a cone preserving map that mostly involve these two concepts when the underlying cone is polyhedral perfect or strictly convex and or smooth or is the cone of all real polynomials of degree not exceeding n that are nonnegative on a closed interval plentiful illustrative examples are provided some open problems are posed at the end 1991 mathematics subject classification primary 15a48 secondary 47b65 47a25 46b42

on linear integral operators with nonnegative kernels abstract let k be an eventually compact linear integral operator on l p 1 p with nonnegative kernel k x y where the underlying measure is totally finite on the domain set when p 1 in considering the equation f k f g for given nonnegative g l p 0 p nelson jr provided necessary and sufficient conditions in terms of the support of g such that a nonnegative solution f l p was attained such conditions led to generalizing some of the graph theoretic ideas associated with the normal form of a nonnegative reducible matrix the purpose of this paper is to show that the analysis by nelson can be enlarged to provide a more complete generalization of the normal form of a nonnegative matrix which can be used to characterize the distinguished eigenvalues of k and k and to describe sets of support for the eigenfunctions and generalized eigenfunctions of both k and k belonging to the spectral radius of k

on nonnegative solvability of linear integral equations abstract let denote a finite measure space and l p 1 p p th summable real valued functions suppose moreover k is an integral operator whose nonnegative kernel k is measurable on and which maps l p into itself while possessing a compact iterate we present necessary and sufficient conditions for the integral operator equation k g to possess a nonnegative solution l p whenever g is a given nontrival and nonnegative element of l p and is any given positive parameter this analysis extends that by victory siam j algebraic discrete methods 6 406 412 1985 for the matrix case

on the generalized nullspace of m matrices and z matrices a proof is given for the preferred basis theorem for the generalized nullspace of a given m matrix the theorem is then generalized and extended to the case of a z matrix

on the collatz wielandt sets associated with a cone preserving map abstract let k be a closed pointed full cone in r n in their treatment of perron frobenius theory for a linear map a preserving k by wielandt s approach barker and schneider introduced four sets namely 1 and 1 the collatz wielandt sets associated with a we determine the greatest lower bound and the least upper bound of these sets some known results for a nonnegative matrix relating its spectral radius to its upper and lower collatz wielandt number are extended to the setting of a cone preserving map applications of our results to the nonnegativity of solutions of linear inequalities associated with a nonnegative matrix are also considered

on the duality operator of a convex cone let c be a convex set in r for each y c the cone of c at y denoted by cone y c is the cone a x y a 0 and x e c if k is a cone in r we shall denote by k its dual cone and by k the lattice of faces of k then the duality operator of k is the mapping dk k k given by dn f span f a cq k properties of the duality operator d k of a closed pointed hill cone k have been studied before in this paper we study da for a general cone k especially m relation to d o c v k where y e k our main result says that for any closed cone k in r the duality operator d k is injective surjective if and only if the duality operator d o u n is injective surjective for each vector y e k k n k in the last part of the paper we obtain some partial results on the problem of constructing a compact convex set c which contains the zero vector such that cone 0 c is equal to a given cone

upper and lower bounds for ranks of matrix expressions using generalized inverses abstract the maximal and minimal ranks of the matrix expression a 1 b 1 x 1 c 1 b 2 x 2 c 2 with respect to x 1 and x 2 are presented as applications the maximal and minimal ranks of a 1 b 1 xc 1 subject to a consistent matrix equation b 2 xc 2 a 2 are also determined in addition the maximal and minimal ranks of the schur complement d ca b with respect to the generalized inverse a of a and their various consequences are also considered

dynamics near an isolated m semi static homoclinic orbit abstract we show that there are infinitely many periodic orbits in any neighborhood of an isolated m semi static orbit homoclinic to an aubry set for time periodic positive lagrangian systems

the influence of the marked reduced graph of a nonnegative matrix on the jordan form and on related properties a survey abstract for a nonnegative matrix p we discuss the relation of its marked reduced graph to that part of the jordan form that is associated with the perron frobenius root to the nonnegativity of the eigenvectors and generalized eigenvectors to the nonnegativity of solutions of linear equations and to the asymptotic growth of powers of the matrix results are often stated in terms of m matrices and standard results on irreducible matrices are assumed we give examples to illustrate the theorems surveyed

completing a matrix when certain entries of its inverse are specified abstract suppose a a 11 a 12 a 21 a 22 is an n n matrix where a 11 has order p q if a is nonsingular let a 1 have the transposed partitioning b 11 b 12 b 21 b 22 first we show that the nullities of a 11 and b 22 are equal the remainder of our paper deals with the following completion problem suppose m 1 m 2 n 1 n 2 are nonnegative integers such that m 1 m 2 n 1 n 2 n 0

on minimality in the partial realization problem abstract a compression algorithm is constructed which allows one to reduce an arbitrary realization of a finite sequence of matrices m 1 m r to a minimal partial realization of m 1 m r furthermore a state space criterion of minimality of a partial realization and a formula for the minimal state space dimension are obtained

geometric conditions for the existence of positive eigenvalues of matrices finite dimensional theorems of perron frobenius type are proved for a e e nn and a nonnegative integer k we let w t a be the cone generated by ai ak l in en we show that a satisfies the perron schaefer condition if and only if the closure wk a of w t a is a pointed cone this theorem is closely related to several known results if k po a the index of the eigenvalue 0 in spec a we prove that a has a positive eigenvalue if and only if w t a is a pointed nonzero cone or equivalently w t a is not a real subspace of en our proofs are elementary and based on a method of birkhoff s we discuss the relation of this method to pringsheim s theorem