Spectral Representation of Analytic Diagonalizable Matrix-valued Functions that Commute with their Derivative

Author: Turcotte D.  

Publisher: Taylor & Francis Ltd

ISSN: 0308-1087

Source: Linear and Multilinear Algebra, Vol.50, Iss.2, 2002-03, pp. : 181-183

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Abstract

In 1981, Goff showed that an analytic n × n self-adjoint matrix-valued function A defined on an open interval I of the real axis that commutes with its derivative must be functionally commutative. We generalize Goff's result by showing that it remains valid for diagonalizable matrices and that we need only require this diagonalizability property on an open subinterval Io of I. We also show that diagonalizability property propagates from the subinterval Io to the whole interval I and we obtain the spectral representation of the function A and its derivatives DjA, j = 0,1,2, .