Free loci of matrix pencils and domains of noncommutative rational functions

Abstract

Consider a monic linear pencil $L(x)=I-A_1{x}_1-\cdots -A_g{x}_g$ 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 $A^{\prime }$ generated by the coefficients of two linear pencils $L$ and $L^{\prime }$, respectively, with equal free loci are isomorphic up to radical. Furthermore, $Z(L)\subseteq Z(L^{\prime })$ if and only if the natural map sending the coefficients of $L^{\prime }$ to the coefficients of $L$ induces a homomorphism $A^{\prime }/\mathrm{rad}{A}^{\prime }\to A/\mathrm{rad}A$. 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.

v4 adds an appendix due independently to Claudio Procesi and Špela Špenko presenting an invariant-theoretic viewpoint of the paper; v2 is the final version that appeared in print

Cited by in corpus (8)

• Geometry of free loci and factorization of noncommutative polynomials
• Bianalytic Maps Between Free Spectrahedra
• Local theory of free noncommutative functions: germs, meromorphic functions and Hermite interpolation
• Bianalytic free maps between spectrahedra and spectraballs
• The free Grothendieck theorem
• Noncommutative polynomials describing convex sets
• Ranks of linear matrix pencils separate simultaneous similarity orbits
• Factorization of noncommutative polynomials and Nullstellensätze for the free algebra