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 $(L)=\{X:\det L(X)=0\}$. In this article it is shown that the algebras $\mathcal{A}$ and $\tilde{\mathcal{A}}$ generated by the coefficients of two linear pencils $L$ and $\tilde{L}$, respectively, with equal free loci are isomorphic up to radical, i.e., $\mathcal{A}/\mathrm{rad}\mathcal{A}\cong \tilde{\mathcal{A}}/\mathrm{rad}\tilde{\mathcal{A}}$. Furthermore, $(L)\subseteq (\tilde{L})$ if and only if the natural map sending the coefficients of $\tilde{L}$ to the coefficients of $L$ induces a homomorphism $\tilde{\mathcal{A}}/\mathrm{rad}\tilde{\mathcal{A}}\to \mathcal{A}/\mathrm{rad}\mathcal{A}$. 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.