Abstract
LetA(ε) andB(ε) be complex valued matrices analytic in ε at the origin.A(ε)≈ p B(ε) ifA(ε) is similar toB(ε) for any |ε|<r,A(ε)≈a B(ε) ifB(ε)=T(ε)A(ε)T −1(ε) andT(ε) is analytic and |T(ε)|≠0 for |ε|<r! In this paper we find a necessary and sufficient conditions onA(ε) andB(ε) such thatA(ε)≈ a B(ε) provided thatA(ε)≈ p B(ε). This problem arises in study of certain ordinary differential equations singular with respect to a parameter ε in the origin and was first stated by Wasow.
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References
F. R. Gantmacher,The Theory of Matrices, Vol. I, Chelsea, New York, 1964.
M. Marcus and H. Minc,A Survey of Matrix Theory and Matrix Inequalities, Prindle, Weber and Schmidt, 1964.
W. Wasow,On holomorphically similar matrices, J. Math. Anal. Appl.4 (1962), 202–206.
W. Wasow,Topics in the theory of linear ordinary, differential equations having singularities with respect to a parameter, IRMA, Universite Louis Pasteur, Strasbourg, 1978.
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Sponsored by the United States Army under Contract No. DAAG29-75-C-0024
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Friedland, S. On pointwise and analytic similarity of matrices. Israel J. Math. 35, 89–108 (1980). https://doi.org/10.1007/BF02760940
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DOI: https://doi.org/10.1007/BF02760940