Free loci of matrix pencils and domains of noncommutative rational functions

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 $A^{\mathrm{\prime }}$ generated by the coefficients of two linear pencils $L$ and $L^{\mathrm{\prime }}$, respectively, with equal free loci are isomorphic up to radical. Furthermore, $Z(L)\subseteq Z(L^{\mathrm{\prime }})$ if and only if the natural map sending the coefficients of $L^{\mathrm{\prime }}$ to the coefficients of $L$ induces a homomorphism $A^{\mathrm{\prime }}\mathrm{/}\mathrm{r}\mathrm{a}\mathrm{d}{A}^{\mathrm{\prime }}\to A\mathrm{/}\mathrm{r}\mathrm{a}\mathrm{d}A$. 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.