Free loci of matrix pencils and domains of noncommutative rational functions

Abstract

Abstract. Consider a monic linear pencil L(x) = I − A 1 x 1 − · · · − A g x g 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 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, i.e., A/ rad A ∼ = A/ rad A. Furthermore, Z (L) ⊆ Z ( L) if and only if the natural map sending the coefficients of L to the coefficients of L induces a homomorphism A/ rad A → 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 A 1 , . . . , A g generate M d (C) as an algebra, then there exist hermitian matrices X 1 , . . . , X g such that i A i ⊗ X i has a simple eigenvalue.The arXiv version of the article contains an appendix presenting an invariant-theoretic viewpoint and is due independently to Claudio Procesi andŠpelaŠpenko.

Bianalytic free maps between spectrahedra and spectraballs

“…Proof. Zalar [Zal17] (see also [HKM13]) establishes this result over the reals, but the proofs work (and are easier) over C; it can also be deduced from the results in [KV17] and [HKV18].…”

“…(3) E is ball-minimal if and only if L re E is minimal. 3 Previously, in [KV17] such pencils were called irreducible.…”

The free Grothendieck theorem

“…Lemma implies , the inverse of , is the unique solution to a proper algebraic polynomial that is ‐affine linear. The ‐affine linearity is reminiscent of realizations of nc rational functions, see . With this similarity to realizations in mind, we generalize the class of rational series (see Definition ) to a slightly larger class of formal power series that we call the hyporational series.…”

“…If , then has a realization; there exist , and such that Classical realization theory has a long and storied history in both mathematics and applied fields. We use definitions and results from , which provides an excellent exposition of realizations of nc rational functions and their domains.…”

“…If is a bijective free polynomial, then necessarily its inverse is an algebraic mapping. If in addition, is rational, then [, Theorem 4.2] implies is a polynomial. In Section 6 we identify, in terms of proper algebraic polynomial mappings, free rational functions among free algebraic mappings and extend, in Theorem , [, Theorem 4.2] to a larger class of free algebraic functions.…”