### Linear Algebra and Matrix Analysis

Linear algebra plays an important role in the solution of many applied problems; it is also a source of challenging theoretical questions in its own right. Techniques from linear algebra are used in areas such as information retrieval, decision making, population dynamics and systems and control theory.

The Linear Algebra group is engaged in a variety of research topics spanning theory and applications. We are interested in matrix stability theory and the use of linear algebra in dynamical systems and control. We are also active in combinatorial matrix theory and spectral graph theory as well as in tropical linear algebra and max algebra. The theory of nonnegative matrices and its applications in dynamical systems and modern communications is a topic of particular interest to the group. Some sample publications are listed below to give a flavour of our activities.

### Group Members

**Present Members** Ollie Mason , Robert Shorten .

**Past Members** Buket Benek Gursoy, Vahid Bokharaie, Ryan Tifenbach, Jane Breen, Felix Goldberg, Steve Kirkland

### Sample Publications

** Combinatorial Matrix Theory and Spectral Graph Theory**

- C. Godsil, S. Kirkland, S. Severini and J. Smith, Number-theoretic Nature of Communication in Quantum Spin Systems , Physical Review Letters 109 (2012), 050502, 4 pages.
- S. Kirkland, Sign Patterns for Eigenmatrices of Nonnegative Matrices , Linear and Multilinear Algebra 59 (2011), 999-1018.
- S. Kirkland and D. Paul, Bipartite Subgraphs and the Signless Laplacian Matrix , Applicable Analysis and Discrete Mathematics 5 (2011), 1-13.
- S. Kirkland, Algebraic Connectivity for Vertex-Deleted Subgraphs , and a Notion of Vertex Centrality , Discrete Mathematics 310 (2010), 911-921.

** Max Algebra**

- B. Benek Gursoy, O. Mason, S. Sergeev. The Analytic Hierarchy Process , Max Algebra and Multi-objective Optimisation, accepted for publication in Linear Algebra and its Applications, 2012.
- B. Benek Gursoy and O. Mason. P1-max and S-max properties and asymptotic stability in the max algebra , Linear Algebra and its Applications, 435(5), pp. 1008-1018, 2011.
- B. Benek Gursoy and O. Mason. Spectral Properties of matrix polynomials over the max algebra , Linear Algebra and its Applications, 435(7), pp. 1626-1636, 2011

** Nonnegative Matrices **

- F. Knorn, O. Mason, R. Shorten, On Linear Co-?positive Lyapunov Functions for Sets of Linear Positive Systems ,

Automatica, Volume 45, Issue 8, pp. 1943?–?1947, 2009 - S. Kirkland, Fastest Expected Time to Mixing for a Markov Chain on a Directed Graph, Linear Algebra and its Applications 433 (2010), 1988-1996.
- R. Brualdi and S. Kirkland, Totally Nonnegative (0,1)-Matrices, Linear Algebra and its Applications 432 (2010), 1650-1662.
- E. Crisostomi, S. Kirkland and R. Shorten, A Google-like Model of Road Network Dynamics and its Application to Regulation and Control , International Journal of Control 84 (2011), 633-651.

** Stability Theory and Dynamics **

- O. Mason. Diagonal Riccati Stability and Positive Time-Delay Systems. Systems and Control Letters, 61 (1). pp. 6-10
- V.S. Bokharaie, O. Mason and F. Wirth. Stability and positivity of equilibria for subhomogeneous cooperative systems , Nonlinear Analysis:Theory Methods and Applications, 74(17), pp. 6416-6426, 2011
- R. Shorten, O. Mason, C. King. An alternative proof of the result of Barker, Berman and Plemmons, and some extensions , Linear Algebra and its Applications, 430(1), pp. 34-40, 2009.