Free loci of matrix pencils and domains of noncommutative rational functions

TL;DR

This paper characterizes the algebraic structure of linear pencils with identical free loci, relates their domains to noncommutative rational functions, and proves a quantum analogue of Kippenhahn's conjecture involving hermitian matrices.

Contribution

It establishes isomorphisms between algebras generated by pencils with equal free loci and characterizes domains of noncommutative rational functions, also proving a quantum version of Kippenhahn's conjecture.

Findings

Algebras generated by pencils with equal free loci are isomorphic up to radical.

Domain inclusion corresponds to algebra homomorphisms between radicals of generated algebras.

Quantum Kippenhahn's conjecture is proved for hermitian matrices generating full matrix algebra.

Abstract

Consider a monic linear pencil $L(x) = I - A_1x_1 - \cdots - A_gx_g$ whose coefficients $A_j$ are $d \times d$ matrices. It is naturally evaluated at $g$-tuples of matrices $X$ using the Kronecker tensor product, which gives rise to its free locus $Z(L) = \{ X: \det L(X) = 0 \}$. In this article it is shown that the algebras $A$ and $A'$ generated by the coefficients of two linear pencils $L$ and $L'$, respectively, with equal free loci are isomorphic up to radical. Furthermore, $Z(L) \subseteq Z(L')$ if and only if the natural map sending the coefficients of $L'$ to the coefficients of $L$ induces a homomorphism $A'/{\rm rad} A' \to A/{\rm rad} A$. 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 $A_1, \dots, A_g$ generate $M_d(\mathbb{C})$ as an algebra, then there exist hermitian matrices $X_1, \dots, X_g$ such that $\sum_i A_i \otimes X_i$ has a simple eigenvalue.