4th Irish Linear Algebra & Matrix Theory Meeting

• When: 9.30 - 4.30 Tuesday 29 April 2025
• Location: T115 Tara Building, Limerick Campus, Mary Immaculate College

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Closing date for registrations is 8 April.

MIC's Department of Mathematics and Computer Studies gratefully acknowledge financial support from the Irish Mathematical Society.

Session One

Patrick Browne, Technological University of the Shannon: Midwest - Chord Diagrams and Weight systems

In this talk, we explore weight systems in knot theory, i.e. linear functionals on chord diagrams. Chord diagrams, while motivated by singular knots, can be viewed as purely combinatorial objects with rich mathematical structure. The significance of weight systems stems from the fundamental result that every Vassiliev knot invariant determines and is determined by a weight system. Moreover, Lie algebras provide a powerful framework for constructing these weight systems.

This presentation will introduce the connection between chord diagrams, weight systems, and Lie theory. We'll explore this interplay as preliminary research that may reveal new insights into both knot theory and combinatorial structures. The talk will be accessible to those without specialized background in knot theory or Lie algebras, focusing on the connections between these objects.

Niall Madden, University of Galway - Enriched FEMs: Stability and Fast Solvers

Finite Element Methods produce linear systems of equations which, when solved, yield numerical solutions to differential equations. The stability and efficiency of the FEMs depend on properties of the system matrix.

In this talk I'll outline a strategy for enriching the finite dimensional space on which the FEM is posed in order to improve accuracy, but the main focus will be on the surreptitious impact this has on the system matrix in terms of stability (in an M-matrix type way) and factorizabilty of the system matrix.

Session Two

Arani Paul, University College Dublin - Code Equivalence and Conductors

Joint work with John Sheekey and Eimear Byrne.

Code Equivalence Problems (CEPs) have been discussed and studied for a long time not only for their importance in cryptography and cyber-security, but also because they connect to different areas of mathematics such as the Graph Isomorphism Problems and the Tensor Isomorphism Problems. This talk is particularly focused on the CEP for vector rank metric codes. Although rank metric codes have not been studied as extensively as Hamming metric codes, it has become an important topic of research in recent decades because of its applications in numerous sectors of modern-day digital technology.

Here we give a brief introduction to CEP in vector rank metric codes and discuss a way to classify all the equivalence classes of codes for given parameters. The key concepts are classical objects from linear and abstract algebra, namely conductors and idealisers. We will discuss practical implementations of classification algorithms, and discuss possible future directions.

Rachel Quinlan, University of Galway - Idempotent Alternating Sign Matrices

Joint work with Cian O'Brien.

An alternating sign matrix (ASM) is a square (0,1,-1)-matrix in which the nonzero entries alternate between 1 and -1 and sum to 1, within each row and column. Permutation matrices are examples of ASMs, and ASMs generalize permutation matrices in several apparently natural but unexpected ways. Every multiplicative group of nonsingular ASMs is a group of permutation matrices, but the set of all ASMs of size n×n also contains multiplicative groups of singular matrices. The identity element E of such a group is an idempotent ASM, it is equal to its own square. In this talk we will discuss some methods for construction of idempotent ASMs, and identify the minimum rank of an idempotent ASM of specified size.

Padraig Ó Catháin, Dublin City University - Matrices with specified automorphisms

Joint work with Santiago Barrera-Acevedo, Heiko Dietrich, and Ronan Egan.

Combinatorial structures such as strongly regular graphs and projective planes are encoded as incidence matrices, which often have interesting linear algebraic properties. Non-existence results are obtained via algebraic arguments. E.g. the Bruck-Ryser-Chowla theorem gives non-obvious necessary conditions for existence of symmetric designs based on equivalence of quadratic forms, and many non-existence results for difference sets boil down to showing that the eigenvalues of an associated matrix must be norms in a suitable number field.

In this talk, I will discuss (constructive) existence of such matrices, under the assumption of a suitable group of automorphisms. This theory is well known for graphs (i.e. symmetric {0,1}-matrices) but rather more subtle when the matrix contains kth roots of unity. In fact, one can construct an explicit basis for the set of all matrices invariant under a given group representation, and construct the eigenvalues of a given invariant matrix in terms of character sums. Time permitting, I will show how to use these methods to build new complex Hadamard matrices.

Session Three

Helena Šmigoc, University College Dublin - Arbitrarily Finely Divisible Stochastic Matrices

Joint work with Priyanka Joshi.

We will consider the class arbitrarily finely divisible stochastic matrices (AFD+-matrices): stochastic matrices that have a stochastic c-th root for infinitely many natural numbers c. This notion generalises the class of embeddable stochastic matrices. In particular, if A is a transition matrix for a Markov process over some time period, then arbitrary finely divisibility of A inside the set of stochastic matrices is the necessary and sufficient condition for the existence of a transition matrix corresponding to this Markov process over infinitesimally short periods.

We will explore the connection between the spectral properties of an AFD+-matrix A and the spectral properties of a limit point L of its stochastic roots. We will demonstrate a construction of a class of AFD+-matrices with a given limit point L from embeddable matrices, and examine specific cases, including 2×2 matrices, 3×3 circulant matrices, and offer a complete characterisation of AFD+-matrices of rank-two.

Badriah Safarji, University of Galway - Rank distributions of matrix representations of graphs over F2

Joint work with Rachel Quinlan and Cian O'Brien.

Over a finite field F, the number of n×n matrices of rank r typically increases as r increases, 0 ≤ r ≤ n. However, over the field of two elements F2, the most frequently occurring rank in Mn(F2) is not n but n-1. The numbers of symmetric F2-matrices of rank n and n-1 coincide if n is odd and differ marginally if n is even. This opens the door to a more thorough investigation of the distribution of the matrix ranks over the field of two elements.

Let Γ be a simple undirected graph. A symmetric matrix M with entries in a field F represents Γ if the off-diagonal entries of M correspond to edges of Γ in the sense that Mij ≠ 0F if and only if xi and xj are adjacent in Γ. The diagonal entries of M are not subject to any constraints, and therefore there are many matrices representing Γ over F. This project aims to identify and characterize simple graphs of order n with more F2-matrix representations of rank n-1 than rank n, a property rare over other finite fields. We restrict our attention to graphs of order n ≥ 3 with an induced subgraph isomorphic to the path Pn-1. This talk will present results on the rank distributions of matrix representations of such graphs over F2.

Bernard Hanzon, University College Cork - Parametrization of Stable Multivariable Systems: Pivot Structures and Numbered Young Diagrams

Joint work with Ralf Peeters and Martine Olivi.

In this presentation we show how stable linear multivariable systems can be parametrized using orthogonal m-upper (m+n)×(m+n) Hessenberg matrices, where m stands for the number of inputs and n for the order of the system. To make sure all systems are covered by the parametrization certain column permutations of the Hessenberg matrix will be utilized. The lower part of the permuted (m-upper) Hessenberg matrix will given the pair [B;A]; where (A;B) is the controllable pair of the system in state space form. Advantages of this approach to parametrization of stable linear systems will be discussed.

Cian O'Brien, Mary Immaculate College - The Bruhat Order for Latin squares

Joint work with Angela Carnevale.

Alternating sign matrices arise naturally as a generalisation of permutation matrices in a number of different contexts. For example, they are the unique minimal lattice extension of the permutation matrices under the Bruhat order. In 2018, Brualdi and Dahl defined alternating sign hypermatrices, a 3-dimensional analogue of alternating sign matrices. Latin squares can be thought of as the 3-dimensional analogue of permutation matrices, since the positions of each of the n symbols in an n×n Latin square correspond to the non-zero entries in some n×n permutation matrix.

We have extended this idea further, by defining the Bruhat order for Latin squares and studying the resulting poset. This talk presents current work relating to this poset, including 3-dimensional analogues of related combinatorial objects, and a lattice extension of the Latin square poset.

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