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linear-recurr

# linear-recurr - Linear Recurrence using matrices Jonathan...

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Linear Recurrence using matrices Jonathan L.F. King University of Florida, Gainesville FL 32611 [email protected] Webpage http://www.math.ufl.edu/ squash/ 26 October, 2011 (at 10:39 ) See also Problems/NumberTheory/fibonacci.latex Notation. Use r for row-equivalence: I.e, K × N matrices M r M 0 IFF we can get from M to M 0 using row operations. Use “ c ” for column- equivalence. Two N × N matrices B & C are similar , or con- jugate to each other, if there exists an invertible matrix U such that U 1 · B · U = C . Use B sim C for the similarity equiv-relation. Call a matrix OTForm s I note ==== [ s 0 0 s ], a dilation ; its action on the plane is simply to scale-uniformly by factor s . Every matrix commutes with I , and so: A dilation is only conjugate to itself. 1 : I.e, for invertible U , necessarily U 1 · s I · U = s I . For T a square matrix ( or a trn from a vectorspace to itself ) and a complex number α , define the subspace E T := vectors v T v = α v . 2 : Saying that α is a T -eigenvalue” is the same as saying that Dim( E T ) > 1.

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