In minisymposium: Polynomial and Rational Eigenvalue Problems
Wed 11:00–11:30, Room AV 91.12In [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.
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