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Hans Schneider

Publications and source records attributed to Hans Schneider.

At least 19 recordsLinked to original sources
arXiv2014

Generalizations of Bounds on the Index of Convergence to Weighted Digraphs

We study sequences of optimal walks of a growing length, in weighted digraphs, or equivalently, sequences of entries of max-algebraic matrix powers with growing exponents. It is known that these sequences are eventually periodic when the digraphs are strongly connected. The transient of such periodicity depends, in general, both on the size of digraph and on the magnitude of the weights. In this paper, we show that some bounds on the indices of periodicity of (unweighted) digraphs, such as the bounds of Wielandt, Dulmage-Mendelsohn, Schwarz, Kim and Gregory-Kirkland-Pullman, apply to the weights of optimal walks when one of their ends is a critical node.

math.CO↗
arXiv2013

On sets of eigenvalues of matrices with prescribed row sums and prescribed graph

Motivated by a work of Boros, Brualdi, Crama and Hoffman, we consider the sets of (i) possible Perron roots of nonnegative matrices with prescribed row sums and associated graph, and (ii) possible eigenvalues of complex matrices with prescribed associated graph and row sums of the moduli of their entries. To characterize the set of Perron roots or possible eigenvalues of matrices in these classes we introduce, following an idea of Al'pin, Elsner and van den Driessche, the concept of row uniform matrix, which is a nonnegative matrix where all nonzero entries in every row are equal. Furthermore, we completely characterize the sets of possible Perron roots of the class of nonnegative matrices and the set of possible eigenvalues of the class of complex matrices under study. Extending known results to the reducible case, we derive new sharp bounds on the set of eigenvalues or Perron roots of matrices when the only information available is the graph of the matrix and the row sums of the moduli of its entries. In the last section of the paper a new constructive proof of the Camion-Hoffman theorem is given.

math.RA↗
arXiv2013

On the max-algebraic core of a nonnegative matrix

The max-algebraic core of a nonnegative matrix is the intersection of column spans of all max-algebraic matrix powers. Here we investigate the action of a matrix on its core. Being closely related to ultimate periodicity of matrix powers, this study leads us to new modifications and geometric characterizations of robust, orbit periodic and weakly stable matrices.

math.RA↗
arXiv2013

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.

math.RA↗
arXiv2012

Matrices that commute with their derivative. On a letter from Schur to Wielandt

We examine when a matrix whose elements are differentiable functions in one variable commutes with its derivative. This problem was discussed in a letter from Issai Schur to Helmut Wielandt written in 1934, which we found in Wielandt's Nachlass. We present this letter and its translation into English. The topic was rediscovered later and partial results were proved. However, there are many subtle observations in Schur's letter which were not obtained in later years. Using an algebraic setting, we put these into perspective and extend them in several directions. We present in detail the relationship between several conditions mentioned in Schur's letter and we focus in particular on the characterization of matrices called Type 1 by Schur. We also present several examples that demonstrate Schur's observations.

math.RA↗
arXiv2012

Z-matrix equations in max algebra, nonnegative linear algebra and other semirings

We study the max-algebraic analogue of equations involving Z-matrices and M-matrices, with an outlook to a more general algebraic setting. We show that these equations can be solved using the Frobenius trace down method in a way similar to that in non-negative linear algebra, characterizing the solvability in terms of supports and access relations. We give a description of the solution set as combination of the least solution and the eigenspace of the matrix, and provide a general algebraic setting in which this result holds.

math.RA↗
arXiv2011

CSR expansions of matrix powers in max algebra

We study the behavior of max-algebraic powers of a reducible nonnegative n by n matrix A. We show that for t>3n^2, the powers A^t can be expanded in max-algebraic powers of the form CS^tR, where C and R are extracted from columns and rows of certain Kleene stars and S is diadonally similar to a Boolean matrix. We study the properties of individual terms and show that all terms, for a given t>3n^2, can be found in O(n^4 log n) operations. We show that the powers have a well-defined ultimate behavior, where certain terms are totally or partially suppressed, thus leading to ultimate CS^tR terms and the corresponding ultimate expansion. We apply this expansion to the question whether {A^ty, t>0} is ultimately linear periodic for each starting vector y, showing that this question can be also answered in O(n^4 log n) time. We give examples illustrating our main results.

math.RA↗
arXiv2010

On commuting matrices in max algebra and in classical nonnegative algebra

This paper studies commuting matrices in max algebra and nonnegative linear algebra. Our starting point is the existence of a common eigenvector, which directly leads to max analogues of some classical results for complex matrices. We also investigate Frobenius normal forms of commuting matrices, particularly when the Perron roots of the components are distinct. For the case of max algebra, we show how the intersection of eigencones of commuting matrices can be described, and we consider connections with Boolean algebra which enables us to prove that two commuting irreducible matrices in max algebra have a common eigennode.

math.RA↗
arXiv2009

On visualisation scaling, subeigenvectors and Kleene stars in max algebra

The purpose of this paper is to investigate the interplay arising between max algebra, convexity and scaling problems. The latter, which have been studied in nonnegative matrix theory, are strongly related to max algebra. One problem is strict visualisation scaling, which means finding, for a given nonnegative matrix A, a diagonal matrix X such that all elements of X^{-1}AX are less than or equal to the maximum cycle geometric mean of A, with strict inequality for the entries which do not lie on critical cycles. In this paper such scalings are described by means of the max-algebraic subeigenvectors and Kleene stars of nonnegative matrices as well as by some concepts of convex geometry.

math.MG↗
arXiv2006

Generators, extremals and bases of max cones

Max cones are max-algebraic analogs of convex cones. In the present paper we develop a theory of generating sets and extremals of max cones in ${\mathbb R}_+^n$. This theory is based on the observation that extremals are minimal elements of max cones under suitable scalings of vectors. We give new proofs of existing results suitably generalizing, restating and refining them. Of these, it is important that any set of generators may be partitioned into the set of extremals and the set of redundant elements. We include results on properties of open and closed cones, on properties of totally dependent sets and on computational bounds for the problem of finding the (essentially unique) basis of a finitely generated cone.

math.RA↗
arXiv2005

Potter, Wielandt, and Drazin on the matrix equation $AB=ωBA$, with some new answers to old questions

In this partly historical and partly research oriented note, we display a page of an unpublished mathematical diary of Helmut Wielandt's for 1951. There he gives a new proof of a theorem due to H. S. A. Potter on the matrix equation $AB = ωBA$, which is related to the $q$-binomial theorem, and asks some further questions, which we answer. We also describe results by M. P. Drazin and others on this equation.

math.RA↗
arXiv2001

Linear Equations over cones and Collatz-Wielandt numbers

Let $K$ be a proper cone in $\IR^n$, let $A$ be an $n\times n$ real matrix that satisfies $AK\subseteq K$, let $b$ be a given vector of $K$, and let $λ$ be a given positive real number. The following two linear equations are considered in this paper: (i)$(λI_n-A)x=b$, $x\in K$, and (ii)$(A-λI_n)x=b$, $x\in K$. We obtain several equivalent conditions for the solvability of the first equation. For the second equation we give an equivalent condition for its solvability in case when $λ>ρ_b (A)$, and we also find a necessary condition when $λ=ρ_b (A)$ and also when $λ< ρ_b(A)$, sufficiently close to $ρ_b(A)$, where $ρ_b (A)$ denotes the local spectral radius of $A$ at $b$. With $λ$ fixed, we also consider the questions of when the set $(A-λI_n)K \bigcap K$ equals $\{0\}$ or $K$, and what the face of $K$ generated by the set is. Then we derive some new results about local spectral radii and Collatz-Wielandt sets (or numbers) associated with a cone-preserving map, and extend a known characterization of $M$-matrices among $Z$-matrices in terms of alternating sequences.

math.RA↗
arXiv2001

Applications of Perron-Frobenius Theory to Population Dynamics

By the use of Perron-Frobenius theory, simple proofs are given of the Fundamental Theorem of Demography and of a theorem of Cushing and Yicang on the net reproductive rate occurring in matrix models of population dynamics. The latter result is further refined with some additional nonnegative matrix theory. When the fertility matrix is scaled by the net reproductive rate, the growth rate of the model is 1. More generally, we show how to achieve a given growth rate for the model by scaling the fertility matrix. Demographic interpretations of the results are given.

math.RA↗
arXiv1998

Hermitian Positive Semidefinite Matrices Whose Entries Are 0 Or 1 in Modulus

We show that a matrix is a Hermitian positive semidefinite matrix whose nonzero entries have modulus 1 if and only if it similar to a direct sum of all $1's$ matrices and a 0 matrix via a unitary monomial similarity. In particular, the only such nonsingular matrix is the identity matrix and the only such irreducible matrix is similar to an all 1's matrix by means of a unitary diagonal similarity. Our results extend earlier results of Jain and Snyder for the case in which the nonzero entries (actually) equal 1. Our methods of proof, which rely on the so called principal submatrix rank property, differ from the approach used by Jain and Snyder.

math.RA↗
arXiv1998

Block LU factorization of M-matrices

It is well known that any nonsingular M-matrix admits an LU factorization into M-matrices (with L and U lower and upper triangular respectively) and any singular M-matrix is permutation similar to an M-matrix which admits an LU factorization into M-matrices. Varga and Cai establish necessary and sufficient conditions for a singular M-matrix (without permutation) to allow an LU factorization with L nonsingular. We generalize these results in two directions. First, we find necessary and sufficient conditions for the existence of an LU factorization of a singular M-matrix where L and U are both permitted to be singular. Second, we establish the minimal block structure that a block LU factorization of a singular M-matrix can have when L and U are M-matrices.

math.RA↗
arXiv1998

On matrices for which norm bounds are attained

Let $\|A\|_{p,q}$ be the norm induced on the matrix $A$ with $n$ rows and $m$ columns by the Hölder $\ell_p$ and $\ell_q$ norms on $R^n$ and $R^m$ (or $C^n$ and $C^m$), respectively. It is easy to find an upper bound for the ratio $\|A\|_{r,s}/\|A\|_{p,q}$. In this paper we study the classes of matrices for which the upper bound is attained. We shall show that for fixed $A$, attainment of the bound depends only on the signs of $r-p$ and $s-q$. Various criteria depending on these signs are obtained. For the special case $p=q=2$, the set of all matrices for which the bound is attained is generated by means of singular value decompositions.

math.RA↗