Workshop on Linear Algebra & Applications

17 October 2011 — Hamilton Institute


Sunday 16 October 2011

19:00 Informal get-together in “Brady's Clock House Pub” (finger food will be served)

Monday 17 October 2011

8:00 Welcome breakfast (at the Hamilton Institute)
9:00 Opening address by R. Shorten and D. Heffernan
Shmuel Friedland: From nonnegative matrices to nonnegative tensors
Abstract In this talk we will discuss a number of generalizations of results on nonnegative matrices to nonnegative tensors as: irreducibility and weak irreducibility, Perron-Frobenius theorem, Collatz-Wielandt characterization, Kingman's inequality, Karlin-Ost and Friedland theorems, tropical spectral radius, diagonal scaling, Friedland-Karlin inequality, nonnegative multilinear forms.
Raphael Loewy: Maximal exponents of polyhedral cones
Abstract Let K be a proper (i.e., closed, pointed, full and convex) cone in Rn. We consider ARn×n which is K-primitive, that is, there exists a positive integer l such that Alx ∈ int K for every 0≠xK. The smallest such l is called the exponent of A, denoted by γ(A).

For a polyhedral cone K, the maximum value of γ(A), taken over all K-primitive matrices A, is denoted by γ(K). Our main result is that for any positive integers m,n, 3 ≤ nm, the maximum value of γ(K), as K runs through all n-dimensional polyhedral cones with m extreme rays, equals

( n - 1 )( m - 1 ) + ½( 1 + (-1)(n-1)m ).

We will consider various uniqueness issues related to the main result as well as its connections to known results.

This talk is based on a joint work with Micha Perles and Bit-Shun Tam.
10:45 Coffee
Thomas J. Laffey: Some relationships between formal power series and nonnegative matrices
Abstract Let σ = (λ1,…,λn) be a list of complex numbers which we aim to realize constructively as the spectrum of a nonnegative matrix. Most constructions available in the literature rely on building matrices related to companion matrices from the polynomial f(x) = (xλ1)…(xλn). Kim, Ormes and Roush (JAMS 2000) showed how certain formal power series related to f(x), which have all coefficients, other than the leading one, negative, can be used in finding constructions over the semiring of polynomials with nonnegative coefficients, while, in joint work, Šmigoc and this author (ELA 17 (2008) 333-342, LAMA 58 (2010), 1053-1059) have used polynomials having all their non-leading coefficients negative, to find realizations when σ has not more than two entries with positive real parts. Beginning with the observation that if λ1,…,λn are all positive, then the Taylor expansion of the nth root of F(t) = (1–λ1t)…(1–λnt) about t=0 has all its non-leading coefficients negative, we present a number of results on the negativity of the coefficients of power series and their applications to nonnegative matrices.
Patrizio Colaneri: Essentially Negative News About Positive Systems
Abstract In this paper the discretisation of switched and non-switched linear positive systems using Padé approximations is considered. Padé approximations to the matrix exponential are sometimes used by control engineers for discretising continuous time systems and for control system design. We observe that this method of approximation is not suited for the discretisation of positive dynamic systems, for two key reasons. First, certain types of Lyapunov stability are not, in general, preserved. Secondly, and more seriously, positivity need not be preserved, even when stability is. Finally we present an alternative approximation to the matrix exponential which preserves positivity, and linear and quadratic stability.

This talk is based on joint work with Steve Kirkland, Annalisa Zappavigna & Robert Shorten
12:45 Lunch
Karl-Heinz Förster: On the Block Numerical Range of Operators in Banach Spaces
Abstract In this talk following topics will be discussed:
  1. The Numerical Range of Operators in Banach Spaces.
  2. The Block Numerical Range of Operators.
  3. The Block Numerical Range of Operator Functions.
  4. The Block Numerical Range of m-monic Perron-Frobenius-Matrix-Polynomials.
Helena Šmigoc: The Symmetric Nonnegative Inverse Eigenvalue Problem
Abstract The question of which lists of complex numbers are the spectra of nonnegative matrices, is known as the nonnegative inverse eigenvalue problem, and the same question posed for symmetric nonnegative matrices is called the symmetric nonnegative inverse eigenvalue problem. In the talk we will present an overview of some recent results on the symmetric nonnegative inverse eigenvalue problem.

Joint work with T. J. Laffey.
15:30 Coffee
Steve Kirkland: Load balancing for Markov chains
Abstract A square matrix T is called stochastic if its entries are nonnegative and its row sums are all equal to one. Stochastic matrices are the centrepiece of the theory of discrete-time, time homogenous Markov chains on a finite state space. If some power of the stochastic matrix T has all positive entries, then there is a unique left eigenvector for T, known as the stationary distribution, to which the iterates of the Markov chain converge, regardless of what the initial distribution for the chain is. Thus, in this setting, the stationary distribution can be thought of as giving the probability that the chain is in a particular state over the long run.

In many applications, the stochastic matrix under consideration is equipped with an underlying combinatorial structure, which can be recorded in a directed graph. Given a stochastic matrix T, how are the entries in the stationary distribution influenced by the structure of the directed graph associated with T? In this talk we investigate a question of that type by finding the minimum value of the maximum entry in the stationary distribution for T, as T ranges over the set of stochastic matrices with a given directed graph. The solution involves techniques from matrix theory, graph theory, and nonlinear programming.
Abraham Berman: Diagonal Stability and Completely Positive Matrices
Abstract In this paper a general notion of common diagonal Lyapunov matrix is formulated for a collection of n×n matrices A1,…,As and polyhedral cones k1,…,ks in Rn. Necessary and sufficient conditions are derived for the existence of a common diagonal Lyapunov matrix in this setting.

This talk is based on joint work with Christopher King & Robert Shorten

19:30 Speakers' Banquet in “The Avenue
Any non-speakers please contact if you would like to attend, the cost is 30 EUR p.p.