Free loci of matrix pencils and domains of noncommutative rational functions

Abstract

Consider a monic linear pencil L(x) =I-A1X1-Ag xg whose coefficients A j are d × d matrices. It is naturally evaluated at g-tuples of matrices X using the Kronecker tensor product, which gives rise to its free locus Ⅎ (L) = {X: det L(X) = 0}. In this article it is shown that the algebras A and à generated by the coefficients of two linear pencils L and L, respectively, with equal free loci are isomorphic up to radical, i.e., A/rad A ≅ Ã/rad Ã. Furthermore, Ⅎ (L) ⊆ Ⅎ (L) if and only if the natural map sending the coefficients of L to the coefficients of L induces a homomorphism Ã/rad à → A/rad A. 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 A1., Ag generate Md (ℂ) as an algebra, then there exist hermitian matrices X1,., Xg such that ∑ Ai ⊗ Xi has a simple eigenvalue.