WESTERN CANADA LINEAR ALGEBRA MEETING Program - Abstracts - Participants University of Regina May 10-11, 2014 Organizing Committee Shaun Fallat, Doug Farenick, Hadi Kharaghani, Steve Kirkland, Peter Lancaster, Michael Tsatsomeros, Pauline van den Driessche Local Organizers: Shaun Fallat and Doug Farenick Funding The WCLAM Organising Committee gratefully acknowledges the generous support for this meeting provided by The Pacific Institute for the Mathematical Sciences, the International Linear Algebra Society, the Department of Mathematics & Statistics, the Faculty of Science, and the President’s Office at the University of Regina. Invited Speakers Joel Friedman, University of British Columbia Roger Horn, University of Utah Mitja Mastnak, Saint Mary’s University Maya Mincheva, Northern Illinois University (ILAS Lecturer). Location CL 312 and CL 313 (Break room) of the Classroom Building, University of Regina 1 Program and Participants 1 Meeting Program (CL 312, Classroom Bldg.) Saturday, May 10, 2014 08:00-08:45 Registration 08:45-09:00 Welcome & Information Chair: Doug Farenick 09:00-09:50 Joel Friedman (invited), Sheaves on Graphs, L2 Betti Numbers, and Applications 10:00-10:25 Colin Garnett, Integrally Normalizable Matrices and Zero-nonzero Patterns 10:30-11:00 Break (CL 313) Chair: Shaun Fallat 11:00-11:25 Daniel May, ABCs of Mutually Unbiased Bases 11:30-11:55 Rajesh Pereira, Old and New Problems for Doubly Stochastic Matrices and their Generalizations 12:00-13:30 Lunch (on your own) Chair: Pauline van den Driessche 13:30-14:20 Maya Mincheva (invited & ILAS Lecture), The Jacobian Matrix and Multistationarity in Biochemical Reaction Network Models 14:30-14:55 Lixing Han, Stenger’s Sinc Matrix Conjecture 15:00-15:30 Break (CL 313) Chair: Steve Kirkland 15:30-15:55 Louis Deaett, Cycles in the Graph of a Positive Semidefinite Matrix with Low Rank 16:00-16:25 Minerva Catral, Representations for the Drazin Inverse of Block Cyclic Matrices 16:30-16:55 Ming Hua Lin, Positive Semidefinite 3-by-3 Block Matrices 17:00-17:25 Jadranka Micic, Generalizations of Levinson’s Inequality for Hilbert Space Operators 18:30- Informal Dinner Greko’s Restaurant, 4424 Albert Street South, Regina 2 WCLAM ’14 Sunday, May 11, 2014 Chair: Michael Tsatsomeros 08:30-08:55 Sarah Plosker, Using Vector Spaces of Matrices to Study Quantum Measurements 09:00-09:50 Mitja Mastnak (invited), Semitransitive Collections of Matrices 10:00-10:25 Alexey Popov, Semigroups of Partial Isometries 10:25-11:00 Break and meeting photo (CL 313) Chair: Peter Lancaster 11:00-11:50 Roger Horn (invited), Simultaneous Unitary Similarity and Congruence 12:00-13:30 Lunch (on your own) Chair: Shaun Fallat 13:30-13:55 Peitro Paparella, Matrix Roots of Irreducible, Imprimitive Nonnegative Matrices 14:00-14:25 Jane Breen, Stationary Vectors of Stochastic Matrices subject to Combinatorial Constraints 14:30-14:55 Tin-Yau Tam, Geometric Means for Positive Definite Matrices and its Generalization 15:00-15:25 Peter Lancaster, Spectral Analysis for Matrix Polynomials with Symmetries 15:30- Closing remarks Posters (CL 313 of the Classroom Building) Michael Cavers and Kris Vasudevan, The Kuramoto Model on Undirected and Directed Graphs Garrett Culos, Application of Sign Patterns that Require Hn Program and Participants 2 3 Abstracts for talks (alphabetical by speaker) Stationary vectors of stochastic matrices subject to combinatorial constraints Jane Breen Given a strongly connected directed graph D, let SD denote the set of all stochastic matrices whose directed graph is a spanning subgraph of D. Can we describe the set of stationary vectors of irreducible members of SD? We will use results from the area of convex polytopes and an association of each matrix with an undirected bipartite graph to tackle this question, and discuss applications of this to the area of Markov chains, as well as some more unusual applications. This talk is based on joint work with Steve Kirkland. Representations for the Drazin inverse of block cyclic matrices Minerva Catral We consider block k-cyclic (k ≥ 2) real or complex matrices A with nonzeros only in blocks Ai,i+1 , for i = 1, . . . , k (mod k). A formula for the Drazin inverse of A is presented in terms of the Drazin inverse of a smaller order product of the nonzero blocks of A, namely Bi = Ai,i+1 . . . Ai−1,i for some i. Bounds on the index of A in terms of the minimum and maximum indices of these Bi are presented, and for (entrywise) nonnegative A, a sign pattern analysis of the Drazin inverse is discussed. This talk is based on joint work with P. van den Driessche (University of Victoria). Poster: The Kuramoto model on undirected and directed graphs Mike Cavers & Kris Vasudevan The Kuramoto model is a mathematical model for the behaviour of a large set of coupled oscillators and has been applied to describe synchronization. We investigate this model on graphs to understand the relationship between the structure of a graph and its effect on the dynamics. We consider classical properties of graphs along with some new definitions. For instance, we study topological centrality and Kirchhoff index of undirected graphs that are derived from the Moore-Penrose pseudo inverse of their Laplacians, and attempt to relate the eigenspectrum of the graphs to study the synchronization behavior. For directed graphs, we consider a normalized digraph Laplacian recently introduced in the literature and referred to as the Diplacian. We explore the possibility of its relevance in understanding the synchronization on directed graphs. Poster: Applications of sign patterns that require Hn Garrett Culos 4 WCLAM ’14 Cycles in the graph of a positive semidefinite matrix with low rank Louis Deaett The pattern of the off-diagonal nonzero entries in an n×n Hermitian matrix can be described by a simple graph. Rosenfeld showed that in the positive semidefinite case, this graph can be connected and triangle-free only when the rank of the matrix is at least n/2. Here we present a new proof of this result by an approach that makes the role of the combinatorics more transparent, and follow this approach further to obtain new results regarding cycles in the graph of a positive semidefinite matrix with sufficiently low rank. In particular, when its rank is sufficiently small, such a matrix must have a 4-cycle in its graph, and a similar bound exists for 5-cycles as well. Moreover, these bounds can be improved under conditions on the maximum degree of a vertex in the graph. Sheaves on graphs, L2 Betti numbers, and applications Joel Friedman Sheaf theory and (co)homology, in the generality developed by Grothendieck et al., seems to hold great promise for applications in discrete mathematics. We shall describe sheaves on graphs and their applications to (1) solving the Hanna Neumann Conjecture, (2) the girth of graphs, and (3) understanding a generalization of the usual notion of linear independence. It is not a priori clear that sheaf theory should have any bearing on the above applications. A fundamental tool is what we call the ”maximum excess” of a sheaf; this can be defined quite simply (as the maximum negative Euler characteristic occurring over all subsheaves of a sheaf), without any (co)homology theory. It is probably fundamental because it is essentially an L2 Betti number of the sheaf. In particular, Warren Dicks has given much shorter version of application (1) using maximum excess alone, strengthening and simplifying our methods using skew group rings. This talk assumes only basic linear algebra and graph theory. Part of the material is joint with Alice Izsak and Lior Silberman. Integrally normalizable matrices and zero-nonzero patterns. Colin Garnett It is well known that diagonal similarity is useful for reducing the number of variables in spectral problems involving n × n zero-nonzero patterns. This is achieved by scaling n − 1 of the nonzero entries to be 1 under the diagonal similarity. I am interested in diagonal similarity of matrices over the integers, in particular when the diagonal similarity maintains the integrality of all of the entries in the matrix and scales n − 1 of the entries to be 1. If every integer matrix with a zero-nonzero pattern is diagonally similar to such an integer matrix with the 1s in the same positions, we call the pattern integrally normalizable. In this talk I will discuss two characterizations of integrally normalizable zero-nonzero patterns. 5 Program and Participants Stengers Sinc matrix conjecture Lixing Han In his paper [Journal of Computational and Applied Mathematics, 86 (1997), pp. 297310], Frank Stenger pointed out that the validity of relevant Sinc methods hinges on the assumption that all eigenvalues of the Sinc matrix I (−1) are located in the open right halfplane. In light of favorable numerical evidence for each I (−1) of order up to n = 513, Stenger conjectured that this is the case for all I (−1) , regardless how large n is. In 2003, Iyad AbuJeib and Thomas Shores established a partial answer to this unsolved problem. In 2013, we gave a complete proof of the conjecture. In this talk, we will present the conjecture and give an outline of the proof Simultaneous unitary similarity and congruence Roger Horn Square complex matrices A and B are unitarily similar if there is a unitary matrix U such that A = U BU ∗ (conjugate transpose); they are unitarily congruent if there is a unitary U such that A = U BU T . One can determine whether a given pair of matrices is unitarily similar with finitely many ordinary arithmetic operations. However, for a long time the situation seemed to be different for unitary congruence. The best known result says that two given matrices are unitarily congruent if and only if three specific pairs of matrices are simultaneously unitarily similar (that is, the same U works for all three pairs). So...how can one determine whether some pairs of matrices are simultaneously unitarily similar? What about simultaneous unitary congruence of some pairs of matrices? Spectral analysis for matrix polynomials with symmetries Peter Lancaster Two lines of attack in the spectral theory of n × n matrix polynomials will be outlined. The first is an algebraic approach based on the notion of isospectral linear systems in Cln (the linearizations) and the second on analysis of associated matrix-valued functions acting on Cn . The first approach concerns real symmetric systems and leads to canonical forms consisting of real matrix triples, and thence to canonical triples. Furthermore, for real selfadjoint systems we describe selfadjoint canonical triples of real matrices and illustrate their properties. It turns out that, in this context, there is a fundamental orthogonality property associated with the spectrum. It will be shown how this can play a role in inverse (spectral) problems, i.e. constructing systems with prescribed spectral properties. LPZ P. Lancaster, U. Prells, I. Zaballa, An orthogonality property for real symmetric matrix polynomials... Operators and Matrices, 7, 2013, 357-379. LZ1 P. Lancaster, I. Zaballa, A review of canonical forms... Operator Theory: Advances and Applications, 218, 2012, 425-443. 6 WCLAM ’14 LZ2 P. Lancaster, I. Zaballa, On the inverse symmetric quadratic eigenvalue problem SIAM J. Matrix Anal. Appl., to appear. Positive semidefinite 3-by-3 block matrices MingHua Lin Positive semidefinite matrices partitioned into 2-by-2 blocks are well studied. Such a partition not only leads to beautiful theoretical results, but also provides powerful techniques for various practical problems. However, an analogous partition to 3-by-3 blocks seems not extensively investigated. In this talk, I shall present several results on positive semidefinite 3-by-3 block matrices. The talk is based on a joint project with P. van den Driessche. Semitransitive collections of matrices Mijta Mastnak We say that a collection C of complex n-by-n matrices is semitransitive, or, more precisely, acts semitransitively on the underlying vector space Cn , if for every pair of nonzero vectors x, y in C there is an element A of C such that either Ax = y or Ay = x. The notion coincides with the notion of transitivity for groups of matrices, but not in general. Topological version of the notion can is defined in the obvious way. Semitransitivity was introduced in 2005 by H. Rosenthal and V. Troitsky who first studied it in the context of WOT-closed algebras of Hilbert space operators. It was later studied in finite and infinite dimensional settings by many authors. A good deal of results were obtained, sometimes in line with initial conjectures but quite often not. In the talk I will review some of the major results in the area. I will finish by briefly discussing recent joint work with J. Bernik, where we relate the notion of semitransitivity to the study of prehomogeneous vector spaces. ABCs of Mutually unbiased bases Daniel May Let V be an n-dimensional complex inner product space. A collection B1 , . . . , Bd of or√ thonormal bases of V is a set of mutually unbiased bases (MUBs) if | hu, u′ i | = 1/ n for all u ∈ Bi , u′ ∈ Bj . We discuss Automorphisms, Bounds and Constructions of MUBs, using results on exponential sums which arise in the study of finite geometry. Generalization of Levinson’s inequality for Hilbert space operators Jadranka Micic Levinson [1] proved the following inequality: If f : (0, 2c) → IR satisfies f ′′′ ≥ 0 and pi , xi , yi , i = 1, 2, . . . , n, are such that pi > 0, P n i=1 pi = 1, 0 ≤ xi ≤ c and x1 + y1 = x2 + y2 = . . . = xn + yn = 2c, then the inequality n X i=1 pi f (xi ) − f (¯ x) ≤ n X i=1 pi f (yi ) − f (¯ y) (2.1) 7 Program and Participants holds, where x ¯= Pn i=1 pi xi and y¯ = Pn i=1 pi yi denote the weighted arithmetic means. Numerous papers have been devoted to generalizations and extensions of Levinson’s result. The aforementioned generalizations of Levinson’s inequality assume that the distribution of the points xi is equal to the distribution of the points yi reflected around the point c ∈ [a, b]. Mercer [2] made a significant improvement by replacing this condition of symmetric distribution with the weaker one that the variances of the two sequences are equal. Recently, Baloch, Peˇcari´c and Praljak [3] gave Levinson’s inequality under Mercer’s assumptions for a new class of functions that extends 3−convex functions. The purpose of this presentation is to consider Levinson’s inequality for self-adjoint operators, positive linear mappings and the family of continuous functions as follows. Let f ∈ C(I) be a real valued functions on an arbitrary interval I in R and c ∈ I ◦ , where I ◦ is the interior of I. We say that f ∈ Kc (I) if there exists a constant A such that the function F (x) = f (x) − A2 x2 is concave on I ∩ (−∞, c] and convex on I ∩ [c, ∞). Moreover, we say • that f ∈ Kc (I) if F is operator concave on I ∩ (−∞, c] and operator convex on I ∩ [c, ∞). Let B(H) be the algebra of all bounded linear operators on a complex Hilbert space H and Bh (H) be the real subspace of all self-adjoint operators on H. We give our main result: Let (X1 , . . . , Xn ) be an n−tuple and (Y1 , . . . , Yk ) be a k−tuple of self-adjoint operators Xi , Yj ∈ Bh (H) with spectra contained in [mx , Mx ] and [my , My ], respectively, such that mx < Mx ≤ c ≤ my < My . Let (Φ1 , . . . , Φn ) be an n−tuple and (ΨP 1 , . . . , Ψk ) be a k−tuple n of positive linear mappings Φi , Ψj : B(H) → B(K), such that i=1 Φi (1H ) = 1K and • Pk i=1 Ψi (1H ) = 1K . Let f ∈ Kc ([mx , My ]). If # # " n " k n k 2 2 X X A X A X 2 2 ≤ C2 := Φi (Xi ) Ψi (Yi ) Φi Xi − Ψi Yi − C1 := 2 2 i=1 i=1 i=1 i=1 then n X i=1 Φi f (Xi ) − f n X i=1 Φi (Xi ) ≤ C1 ≤ C2 ≤ k X i=1 Ψi f (Yi ) − f k X i=1 Ψi (Yi ) . (2.2) Further, applying [4] we obtain Levinson’s operator inequality (2.2) for f ∈ Kc ([mx , My ]) with conditions on the spectra of operators. [1] N. Levinson, Generalization of an inequality of Ky Fan, J. Math. Anal. Appl. 8 (1964), 133-134. [2] A. McD. Mercer, Short proof of Jensen’s and Levinson’s inequalities, Math. Gazette 94 (2010), 492–495. [3] I. A. Baloch, J. Peˇcari´c and M. Praljak, Generalization of Levinson’s inequality, J. Math. Inequal., acceptted for publication [4] J.Mi´ci´c, Z.Pavi´c and J.Peˇcari´c, Jensen’s inequality for operators without operator convexity, Linear Algebra Appl. 434 (2011), 1228–1237. 8 WCLAM ’14 The Jacobian matrix and multistationarity in biochemical reaction network models Maya Mincheva Understanding the dynamics of interactions in complex biochemical networks is an important problem in modern cell biology. Biochemical reaction networks are modeled by large nonlinear dynamical systems with many unknown parameters, which complicates their numerical analysis. Important properties, such as the potential of a biochemical reaction network to display multistationarity (the existence of multiple equilibria) can be determined by the network’s structure. In the first part of the talk we will discuss a graph-theoretic condition for multistationarity which includes the positive cycle condition as a special case. This is an interesting application of the Jacobian matrix, which can be associated with the bipartite graph of a mass-action biochemical network. In the second part of the talk we will discuss a model of the MAPK network which is a principal component of many intracellular signaling modules. Two simple parameter inequalities will be presented. If the first inequality is satisfied, the existence of multistationarity is guaranteed. On the other hand, if the second inequality is satisfied, the uniqueness of an equilibrium is guaranteed. Both inequalities are obtained using degree theory arguments, where the degree of a nonlinear function is defined using the determinant of its Jacobian matrix. Matrix roots of irreducible, imprimitive nonnegative matrices Peitro Paparella In this talk, using tools from combinatorial matrix theory, elementary number theory, and classical matrix function theory, we classify the matrix roots of irreducible, imprimitive, nonnegative matrices. Old and new problems for doubly stochastic matrices and their generalizations. Rajesh Pereira We discuss some problems and results on Doubly Stochastic Matrices including our recent solution of the n = 4 case of the Perfect-Mirsky conjecture which gives the minimal region in the complex plane containing the eigenvalues of the four by four doubly stochastic matrices. We look at the theory of diagonal scalings of complex positive definite matrices to doubly quasi-stochastic matrices. We formulate a new unsolved variant of the van der Waerden conjecture and explain its connections to quantum entanglement. Coauthors: Joanna Boneng, Jeremy Levick, David Kribs. 9 Program and Participants Using vector spaces of matrices to study quantum measurements Sarah Plosker A quantum measurement on a finite set is determined by positive semidefinite matrices that can viewed as matrix-valued probabilities. The linear span of these matrices is called an operator system. In this lecture I will explain how operator systems affiliated with quantum measurements can be used to characterize quantum measurements with a certain extremal property called cleanness. Semigroups of partial isometries Alexey Popov A partial isometry is a matrix (or, more generally, an operator on a Hilbert space) which is an isometry when restricted to the orthogonal complement of its kernel; this is a generalization of a unitary matrix. In this talk, we will discuss multiplicative semigroups of partial isometries. Unlike products of unitaries, product of two partial isometries is not necessarily a partial isometry, this happens only when the two partial isometries are compatible in a precise sense. We will outline a general structure theory of semigroups of partial isometries. Also, we will discuss semigroups which are similar to semigroups of partial isometries. Our principal result here is: an irreducible norm closed matrix semigroup is similar to a semigroup of partial isometries if and only if (a) the norms of the members of the semigroup are uniformly bounded above and below and (b) its idempotents commute. This extends a well-known result about bounded groups. We will also indicate further directions of research. Geometric means for positive definite matrices and its generalization Tin-Yau Lam We discuss some results of Bhatia et al that involve geometric means 1/2 1/2 −1/2 −1/2 A# B = A A BA A1/2 1/2 of two n × n positive definite matrices A and B. The geometric means lies on the geodesic joining A and B. We then consider their extensions in the context of symmetric space of the noncompact type. Some partial order relation is given. 10 3 WCLAM ’14 Participants Vijay Agasthian, University of Regina, [email protected] Ruhi Ahmadi, University of Regina, [email protected] Jane Breen, University of Manitoba, [email protected] Sarah Carnochan Naqvi, University of Regina, [email protected] Minerva Catral, Xavier University, [email protected] Michael Cavers, University of Calgary, [email protected] Robert Cohen, Western State Colorado University, [email protected] Garrett Culos, University of Victoria, [email protected] Louis Deaett, Quinnipiac University, [email protected] Shaun Fallat, University of Regina, [email protected] Doug Farenick, University of Regina, [email protected] Joel Friedman, University of British Columbia, [email protected] Colin Garnett, Black Hills State University, [email protected] Chun-Hua Guo, University of Regina, [email protected] Lixing Han, University of Michigan - Flint, [email protected] Roger Horn, University of Utah, [email protected] Steve Kirkland, University of Manitoba, [email protected] Peter Lancaster, University of Calgary, [email protected] MingHua Lin, University of Victoria, [email protected] Mijta Mastnak, Saint Mary’s University, [email protected] Daniel May, Black Hills State University, [email protected] Karen Meagher, University of Regina, [email protected] Jadranka Micic, University of Zagreb, [email protected] Maya Mincheva, Nothern Illinois University, [email protected] Dale Olesky, University of Victoria, [email protected] Peitro Paparella, College of William and Mary, [email protected] Rajesh Pereira, University of Guelph, [email protected] Sarah Plosker, Brandon University, [email protected] Alexey Popov, University of Waterloo, [email protected] Alison Purdy, University of Regina, [email protected] Abolgahsem Soltani, University of Regina, [email protected] Tin-Yau Tam, Auburn University, [email protected] Michael Tsatsomeros, Washington State University, [email protected] Pauline van den Driessche, University of Victoria, [email protected] Kris Vasudevan, University of Calgary, [email protected]