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- W3100687907 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." @default.
- W3100687907 created "2020-11-23" @default.
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- W3100687907 date "2017-01-01" @default.
- W3100687907 modified "2023-09-26" @default.
- W3100687907 title "Free loci of matrix pencils and domains of noncommutative rational functions" @default.
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- W3100687907 doi "https://doi.org/10.4171/cmh/408" @default.
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