Free loci of matrix pencils and domains of noncommutative rational functions

  • Igor Klep

    The University of Auckland, New Zealand
  • Jurij Volčič

    The University of Auckland, New Zealand

Abstract

Consider a monic linear pencil whose coefficients are matrices. It is naturally evaluated at -tuples of matrices using the Kronecker tensor product, which gives rise to its free locus . In this article it is shown that the algebras and generated by the coefficients of two linear pencils and , respectively, with equal free loci are isomorphic up to radical, i.e., . Furthermore, if and only if the natural map sending the coefficients of to the coefficients of induces a homomorphism . Since linear pencils are a key ingredient in studying noncommutative rational functions via realization theory, the above results lead to a characterization of all noncommutative rational functions with a given domain. Finally, a quantum version of Kippenhahn's conjecture on linear pencils is formulated and proved: if hermitian matrices generate as an algebra, then there exist hermitian matrices such that has a simple eigenvalue.

Cite this article

Igor Klep, Jurij Volčič, Free loci of matrix pencils and domains of noncommutative rational functions. Comment. Math. Helv. 92 (2017), no. 1, pp. 105–130

DOI 10.4171/CMH/408