Location
Cupples I Room 113
Start Date
7-19-2016 3:30 PM
End Date
19-7-2016 3:50 PM
Description
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 $Z(L)=\{X:\det L(X)=0\}$. Our main result is the following: $Z(L)\subseteq Z(L')$ if and only if the natural map sending the coefficients of $L'$ to the coefficients of $L$ induces a homomorphism $A'/\operatorname{rad}A'\to A/\operatorname{rad}A$. Since linear pencils are a key ingredient in studying noncommutative rational functions via linear systems realization theory, the above result leads to a characterization of all noncommutative rational functions with a given domain. Finally, an answer to a quantum version of Kippenhahn's conjecture on linear pencils will be given: if hermitian matrices $A_1,\dots,A_g$ generate $M_d(\C)$ as an algebra, then there exist hermitian matrices $X_1,\dots,X_g$ such that $\sum_iA_i\otimes X_i$ has a simple eigenvalue. The talk is based on the joint work with I. Klep.
Free loci of matrix pencils and domains of noncommutative rational functions
Cupples I Room 113
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 $Z(L)=\{X:\det L(X)=0\}$. Our main result is the following: $Z(L)\subseteq Z(L')$ if and only if the natural map sending the coefficients of $L'$ to the coefficients of $L$ induces a homomorphism $A'/\operatorname{rad}A'\to A/\operatorname{rad}A$. Since linear pencils are a key ingredient in studying noncommutative rational functions via linear systems realization theory, the above result leads to a characterization of all noncommutative rational functions with a given domain. Finally, an answer to a quantum version of Kippenhahn's conjecture on linear pencils will be given: if hermitian matrices $A_1,\dots,A_g$ generate $M_d(\C)$ as an algebra, then there exist hermitian matrices $X_1,\dots,X_g$ such that $\sum_iA_i\otimes X_i$ has a simple eigenvalue. The talk is based on the joint work with I. Klep.