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Consider a monic linear pencil $L(x) = I - A_1x_1 - \cdots - A_gx_g$ whosecoefficients $A_j$ are $d \times d$ matrices. It is naturally evaluated at$g$-tuples of matrices $X$ using the Kronecker tensor product, which gives riseto its free locus $Z(L) = \{ X: \det L(X) = 0 \}$. In this article it is shownthat the algebras $A$ and $A'$ generated by the coefficients of two linearpencils $L$ and $L'$, respectively, with equal free loci are isomorphic up toradical. Furthermore, $Z(L) \subseteq Z(L')$ if and only if the natural mapsending the coefficients of $L'$ to the coefficients of $L$ induces ahomomorphism $A'/{\rm rad} A' \to A/{\rm rad} A$. Since linear pencils are akey ingredient in studying noncommutative rational functions via realizationtheory, the above results lead to a characterization of all noncommutativerational functions with a given domain. Finally, a quantum version ofKippenhahn's conjecture on linear pencils is formulated and proved: ifhermitian 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.

Free loci of matrix pencils and domains of noncommutative rational functions