1. A linear homogenous recurrence relation of degree 2 with constant coefficients is a recurrencerelation of the forman = Adn-1+ Ban 2, Vn 2 2,(1)where A and B are fixed real numbers.(a) Show that this relation can be rewritten in a matrix form as follows()-an-1= M(2)where M is a 2 x 2 matrix (that you need to determine).(b) Show by induction thata1= M"(c) We turn now our attention to the Fibonacci sequence (a particular case of (1)) given by( Fo = 0, Fı =1Fn-1+ Fn-2(3)(d) Write the value of the matrix M for the recurrence relation (3) and establish that(:)-F, = (0 1) M" ()(4)(e) In order to determine a closed formula for F, we need to calculate the value of M". Forthis purpose we should diagonalize the matrix M, i.e. we need to find two matrices Pand D such thatM = PDP',(5)where D is a diagonal matrix with its diagonal entries being the eigenvalues of M. Findmatrices P and D such that equation (5) is satisfied.(f) Using equation (5), show thatM = PD"P-1(g) Show that if the diagonal matrix D =then D" =

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1. A linear homogenous recurrence relation of degree 2 with constant coefficients is a recurrence relation of the form an = Aan-1+ Ban 2. , Vn 2 2, (1) where A and B are fixed real numbers. (a) Show that this relation can be rewritten in a matrix form as follows an-1 M (2) an-2 where M is a 2 x 2 matrix (that you need to determine). (b) Show by induction that an

1. A linear homogenous recurrence relation of degree 2 with constant coefficients is a recurrence relation of the form an = Aan-1+ Ban 2. , Vn 2 2, (1) where A and B are fixed real numbers. (a) Show that this relation can be rewritten in a matrix form as follows an-1 M (2) an-2 where M is a 2 x 2 matrix (that you need to determine). (b) Show by induction that an

Advanced Engineering Mathematics

10th Edition

ISBN:9780470458365

Author:Erwin Kreyszig

Publisher:Erwin Kreyszig

Chapter2: Second-order Linear Odes

Section: Chapter Questions

Problem 1RQ

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