On pointwise and analytic similarity of matrices
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.
KeywordsTest Point Characteristic Polynomial Invariant Polynomial Elementary Divisor Linear Ordinary Differential Equation
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