Simultaneous diagonalization of matrices parametrized by a projective algebraic curve
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.
scientific article; zbMATH DE number 4093661