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

20th Conference of the International Linear Algebra Society (ILAS)

20th ILAS Conference

In minisymposium: Polynomial and Rational Eigenvalue Problems

Wed 11:00–11:30, Room AV 91.12
Some Remarks on Matrix Polynomials in nonstandard bases and Linearizations
Philip Saltenberger (TU Braunschweig)
Joint work with Heike Fassbender (TU Braunschweig)

In [1] the vector space $\mathbb{L}_1(P)$ of matrix pencils corresponding to a matrix polynomial $P(\lambda)$ in the standard monomial basis is introduced. Its elements may be regarded as generalizations of the Frobenius companion pencils identifying $\mathbb{L}_1$ to be a particularly suitable arena to look for (structured) linearizations of $P$. The ideas underlying the construction of $\mathbb{L}_1$ were succesfully further developed in the context of other nonstandard polynomial bases (i.e. the Bernstein basis [2] or the Lagrange basis). In this talk we adopt them to matrix polynomials $P(\lambda)$ in Chebyshev basis. Following the ideas of [1] a vector space $\mathbb{T}_1(P)$ for matrix polynomials in Chebyshev basis is presented that inherits many attractive properties from $\mathbb{L}_1$. As in [4] we show how the elements in $\mathbb{T}_1$ may be neatly characterized and that $\mathbb{T}_1$ always contains a dense subset of (strong) linearizations for $P$. Furthermore, we deduce an easy to check linearization condition based on the results of [3], present how eigenvectors for $P$ may be easily recovered from its linearizations and show how blocksymmetric pencils in $\mathbb{T}_1$ may be constructed. In addition, we reveal a close structural resemblance between $\mathbb{T}_1(P)$ and the vector space $\mathbb{L}_1(P)$ that naturally shows up and demonstrate that this structural conformity seems to be a common ground of vector spaces that are created in a similar fashion for other polynomial bases.

  1. D. S. Mackey, N. Mackey, C. Mehl and V. Mehrmann, Vector spaces of linearizations for matrix polynomials, SIAM J. Matrix Anal. Appl., 28 (2006), pp. 971–1004.
  2. D. S. Mackey and V. Perovic, Linearizations of matrix polynomials in Bernstein basis, MIMS EPrint 2014.29, Manchester Institute for Mathematical Sciences, The University of Manchester.
  3. A. Amiraslani, R. Corless and P. Lancaster, Linearizations of matrix polynomials expressed in polynomial bases, IMA Journal of Numerical Analysis, 29 (2009), pp. 141–157.
  4. H. Fassbender and P. Saltenberger, Some Notes on the linearization of matrix polynomials in standard and Tschebyscheff basis, submitted.