## 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 Ⅎ (L) = {X: det L(X) = 0}. In this article it is shown that the algebras A and Ã 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 Ã → 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 Md (ℂ) as an algebra, then there exist hermitian matrices X_{1},., X_{g} such that ∑ Ai ⊗ Xi has a simple eigenvalue.

Original language | English |
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Pages (from-to) | 105-130 |

Number of pages | 26 |

Journal | Commentarii Mathematici Helvetici |

Volume | 92 |

Issue number | 1 |

DOIs | |

State | Published - 1 Jan 2017 |

Externally published | Yes |

## Keywords

- Free locus
- Hyperbolic polynomial
- Kippenhahn's conjecture
- Linear pencil
- Noncommutative rational function
- Real algebraic geometry
- Realization theory

## ASJC Scopus subject areas

- General Mathematics