Simultaneous diagonalization of matrices parametrized by a projective algebraic curve
type: Item
instance of: scholarly article
title: Simultaneous diagonalization of matrices parametrized by a projective algebraic curve
zbMATH Open document ID: 0668.15011
DOI: 10.1016/0024-3795(89)90400-X
author: William A. Adkins
published in: Linear Algebra and its Applications
publication date: 1989-01-01T00:00:00Z
review text: Let A(z) be an \(n\times n\) matrix with complex polynomial entries. A theorem of \textit{S. Friedland} [ibid. 36, 103-109 (1981; Zbl 0452.15003)] gives sufficient conditions for the existence of a constant matrix T such that \(T^{-1}A(z)T\) is diagonal for all z, namely: (1) for each fixed z, A(z) is diagonalizable over \({\mathbb{C}}\); and (2) if \(z_ 0\in {\mathbb{C}}\), and \(\mu_ 1(z)\) and \(\mu_ 2(z)\) are distinct eigenvalues of A(z) for z in some neighbourhood of \(z_ 0\), then \((\mu_ 1(z)-\mu_ 2(z))/(z- z_ 0)\) does not tend to 0 as \(z\to z_ 0\). This generalized a well- known theorem of \textit{T. Motzkin} and \textit{O. Taussky} on diagonalizing pencils of matrices [Trans. Amer. Math. Soc. 80, 387-401 (1955; Zbl 0067.254)]. In the present paper the author generalizes Friedland's theorem to the situation where \({\mathbb{C}}\) is replaced by an arbitrary algebraically closed field, \({\mathbb{C}}[z]\) is replaced by the field k(X) of rational functions on a nonsingular projective algebraic curve, and (1) and (2) are interpreted in a natural way.
Mathematics Subject Classification ID: 15A21
Mathematics Subject Classification ID: 15A54
zbMATH DE Number: 4093661
zbMATH Keywords: polynomial matrix
zbMATH Keywords: diagonalizable
zbMATH Keywords: rational functions
zbMATH Keywords: nonsingular projective algebraic curve
MaRDI profile type: Publication
cites work: A generalization of the Motzkin-Taussky theorem
cites work: Q4143433
cites work: Q3220672
cites work: Pairs of Matrices With Property L. II
cites work: Q5799167
cites work: Q5555887
description: scientific article; zbMATH DE number 4093661