Download e-book for kindle: Deformations of Nilpotent Matrices over Rings and Reduction by Donald G. Babbitt

By Donald G. Babbitt

ISBN-10: 0821823264

ISBN-13: 9780821823262

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Extra resources for Deformations of Nilpotent Matrices over Rings and Reduction of Analytic Families of Meromorphic Differential Equations

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F 3 * where the f. e C[T] 0 are linearly independent over C and f. (t ) = 0 for all j . It follows from the linear independendence of the f. (t )) for all t is invertible. It follows if and only if L. e S 1 1 < j < q. It is sufficient to prove that the L. same C-linear span as the L (m 4 0). e T for (1 < j < q) have the The inclusion in one direction is clear from the relations L = Z f. L. l

Completion of the proof of Theorem 3. Suppose L(t) / L + S~ for some t. Then by Lemma k, there are m 4 o for which L / S", and we ~ ' m r choose v to be a minimal element among these; L € S by Lemma k. Let pep and let g,LT be as in (3). Lemma 5 and Lemma 6 (with h = g, M = L, P = L f ) now imply that L e IXyEp ]. Since p was arbitrary, Lemma 7 ijnplies that L v e [L ,1L]. As 0 ^ L e S, we have a contradiction to (2b), The theorem is completely proved. Remark. If R = C, then all eigenvalues of ad H in E /[L ,E ] are necessarily > 0.

A family neighborhood of t Note that by the lemma, t »-» U(t) of submodules of VL defined in a is said to be regular if it has the properties (a) and (b) of Lemma 1, and if in addition, i(t) is constant for all t in some neighborhood of t . Proposition 3. Suppose t H U(t) is a regular family of submodules of DEFORMATIONS OF NILPOTENT MATRICES 35 V R . Let U(t) be the closure (cf. § 2 A ) of U(t) in V R . Then t H U(t) is a regular family. Proof. Let i(t) = s for all t e N; here and in what follows N is a sufficiently small neighborhood of t which may not remain the same from one context to another.

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Deformations of Nilpotent Matrices over Rings and Reduction of Analytic Families of Meromorphic Differential Equations by Donald G. Babbitt


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