# SIAM AN21 Minisymposium on Bohemian Matrices and Applications

The two-part minisymposium Bohemian Matrices and Applications, organized by Rob Corless and I, took place at the SIAM Annual Meeting, July 22 and 23, 2021. This page makes available slides from some of the talks.

The minisymposium followed a two-part minisymposium on Bohemian matrices at the 2019 ICIAM meeting in Valencia and a 3-day workshop on Bohemian matrices in Manchester in 2018.

For more on Bohemian matrices see the Bohemian matrices website.

Minisymposium description: Bohemian matrices are matrices with entries drawn from a fixed discrete set of small integers (or some other discrete set). The term is a contraction of BOunded HEight Matrix of Integers. Such matrices arise in many applications, and include $(0,1)$ graph incidence matrices and $(-1,1)$ Bernoulli matrices. The questions of interest range from identifying structures in the spectra of particular classes of Bohemian matrix to searching for most ill conditioned matrices within a class, and applications include stress-testing algorithms and software. This minisymposium will report recent theoretical and computational progress as well as open questions.

Putting Skew-Symmetric Tridiagonal Bohemians on the Calendar. Robert M. Corless, Western University, Canada. Abstract. Rob did not use slides but gave his talk using this paper and this Maple worksheet.

Determinants of Normalized Bohemian Upper Hessenberg Matrices. Massimiliano Fasi, Örebro University, Sweden; Jishe Feng, Longdong University, China; Gian Maria Negri Porzio, University of Manchester, United Kingdom. Abstract. Slides.

Experiments on Upper Hessenberg and Toeplitz Bohemians. Eunice Chan, Western University, Canada. Abstract. Slides.

Eigenvalues of Magic Squares and Related Bohemian Matrices. Hariprasad Manjunath Hegde, Indian Institute of Science, Bengaluru, India. Abstract. Slides.

Calculating the 3D Kings Multiplicity Constant. Nicholas Cohen and Neil Calkin, Clemson University, U.S. Abstract. Slides.

Bohemian Inners Inverses: A First Step Toward Bohemian Generalized Inverses. Laureano Gonzalez-Vega, Universidad de Cantabria, Spain; Juan Rafael Sendra, Universidad Alcalá de Henares, Spain; Juana Sendra Pons, Universidad Politécnica de Madrid, Spain. Abstract. Slides.

Recent Progress in the Rational Factorisation of Integer Matrices. Matthew Lettington, Cardiff University, United Kingdom. Abstract. Slides.

Which Columns are Independent? Why does Row Rank = Column Rank? Gilbert Strang, Massachusetts Institute of Technology, U.S. Abstract. Slides.

Bohemian Matrices: the Symbolic Computation Approach. Juana Sendra, Universidad Autónoma de Madrid, Spain; Laureano González-Vega, Universidad de Estudios Financieros en Madrid, Spain; Juan Rafael Sendra, Universidad Alcalá de Henares, Spain. Abstract. Slides.

# Bohemian Matrices in Manchester

Bohemian matrices are families of matrices in which the entries are drawn from a fixed discrete set of small integers (or some other discrete set). The term is a contraction of BOunded HEight Matrix of Integers and was coined by Rob Corless and Steven Thornton of the University of Western Ontario. Such matrices arise in many situations:

• adjacency matrices of graphs have entries from $\{0, 1\}$;
• Bernoulli matrices, which occur in compressed sensing, have entries from $\{-1,1\}$;
• Hadamard matrices have entries from $\{-1,1\}$ and orthogonal columns; and
• matrices with elements from $\{-1, 0, 1\}$ provide worst case growth factors for Gaussian elimination with partial pivoting and yield the most ill conditioned triangular matrices with elements bounded by $1$.

Rob’s group have done extensive computations of eigenvalues and characteristic polynomials of Bohemian matrices, which have led to interesting observations and conjectures. Many beautiful visualizations are collected on the website http://www.bohemianmatrices.com as well as on the Bohemian Matrices Twitter feed.

In June 2018, Rob and I organized a 3-day workshop Bohemian Matrices and Applications, bringing together 16 people with an interest in the subject from a variety of backgrounds. The introductory talks by Rob, Steven, and I were videod (and are embedded below), and the slides from most of the talks are available on the conference website.

We scheduled plenty of time for discussion and working together. New collaborations were started, several open problems were solved and numerous new questions were posed.

The workshop has led to various strands of ongoing work. Steven has created the Characteristic Polynomial Database, which contains more than $10^9$ polynomials from more than $10^{12}$ Bohemian matrices and has led to a number of conjectures concerning matches of properties to sequences at the On-Line Encyclopedia of Integer Sequences. Three recent outputs are

Sponsorship of the workshop by the Heilbronn Institute for Mathematical Research, the Royal Society and the School of Mathematics at the University of Manchester, as well as support from Canada for some of the Canadian participants, is gratefully acknowledged.