ILAS2016 — 11–15 July 2016 — KU Leuven, Belgium

## 20th ILAS Conference

In minisymposium: Combinatorial Matrix Theory

Tue 16:00–16:30, Room AV 04.17
Kevin N. Vander Meulen (Redeemer University College)
Joint work with Brydon Eastman (McMaster University)

The Frobenius companion matrix is well-known. In 2003, Fiedler [2] described a larger class of sparse companion matrices, some of which are pentadiagonal. In 2014, it was discovered that the class of sparse companion matrices included an even larger set of matrices. In particular, in [1], the class of sparse companion matrices was completely characterized in terms of unit Hessenberg matrices.

In this presentation, we describe the sparse companion matrices that have the lowest bandwidth, that is, we characterize which sparse companion matrices are permutationally similar to a pentadiagonal matrix. The characterization allows us to determine which of the Fiedler companion matrices are permutationally similar to a pentadiagonal matrix. We also describe how to factor a sparse companion matrix into a product of Fiedler factors, permutation matrices and elementary matrices. This work recently appeared in [3].

1. B. Eastman, I.-J. Kim, B. Shader and K.N. Vander Meulen, Companion matrix patterns, Linear Algebra and its Applications 463 (2014), pp. 255–272.
2. M. Fiedler, A note on companion matrices, Linear Algebra and its Applications 372 (2003), pp. 325–331.
3. B. Eastman, and K.N. Vander Meulen, Pentadiagonal Companion Matrices, Special Matrices 4 (2016), pp 13-30.