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Unformatted text preview: Linear Recurrence using matrices Jonathan L.F. King University of Florida, Gainesville FL 32611 email@example.com 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 IFF we can get from M to M 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 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 , define the subspace E T , := vectors v T v = v . 2 : Saying that is a T-eigenvalue is the same as saying that Dim( E T , ) > 1....
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