Israel Journal of Mathematics

, Volume 35, Issue 1–2, pp 89–108 | Cite as

On pointwise and analytic similarity of matrices

  • Shmuel Friedland


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.


Test Point Characteristic Polynomial Invariant Polynomial Elementary Divisor Linear Ordinary Differential Equation 
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  1. 1.
    F. R. Gantmacher,The Theory of Matrices, Vol. I, Chelsea, New York, 1964.Google Scholar
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    M. Marcus and H. Minc,A Survey of Matrix Theory and Matrix Inequalities, Prindle, Weber and Schmidt, 1964.Google Scholar
  3. 3.
    W. Wasow,On holomorphically similar matrices, J. Math. Anal. Appl.4 (1962), 202–206.MATHCrossRefMathSciNetGoogle Scholar
  4. 4.
    W. Wasow,Topics in the theory of linear ordinary, differential equations having singularities with respect to a parameter, IRMA, Universite Louis Pasteur, Strasbourg, 1978.Google Scholar

Copyright information

© The Weizmann Science Press of Israel 1980

Authors and Affiliations

  • Shmuel Friedland
    • 1
  1. 1.Mathematics Research CenterUniversity of WisconsinMadisonU.S.A.

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