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Journal article
Free loci of matrix pencils and domains of noncommutative rational functions
Commentarii mathematici Helvetici, v 92(1), pp 105-130
01 Jan 2017
Abstract
Consider a monic linear pencil L(x) = I - A(1)x(1) -center dot center dot center dot- A(g)x(g) whose coefficients A(j) are d x 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(L) = {X : det L(X) = 0}. In this article it is shown that the algebras A and (A) over tilde generated by the coefficients of two linear pencils L and (L) over tilde, respectively, with equal free loci are isomorphic up to radical, i.e., A/rad A congruent to (A) over tilde/ rad (A) over tilde. Furthermore, L(L) subset of L ((L) over tilde) if and only if the natural map sending the coefficients of (L) over tilde to the coefficients of L induces a homomorphism (A) over tilde /rad (A) over tilde -> A/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),..., A(g) generate M-d (C) as an algebra, then there exist hermitian matrices X-1 ,..., X-g such that Sigma(i) A(i) circle times X-i has a simple eigenvalue.
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Details
- Title
- Free loci of matrix pencils and domains of noncommutative rational functions
- Creators
- Igor Klep - University of AucklandJurij Volcic - University of Auckland
- Publication Details
- Commentarii mathematici Helvetici, v 92(1), pp 105-130
- Publisher
- European Mathematical Soc
- Number of pages
- 26
- Grant note
- University of Auckland Doctoral Scholarship Marsden Fund Council of the Royal Society of New Zealand; Royal Society of New Zealand; Marsden Fund (NZ) P1-0222; L1-4292; L1-6722 / Slovenian Research Agency; Slovenian Research Agency - Slovenia
- Resource Type
- Journal article
- Language
- English
- Academic Unit
- Mathematics
- Web of Science ID
- WOS:000397060600005
- Scopus ID
- 2-s2.0-85015994966
- Other Identifier
- 991021861881304721
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- Mathematics