Complex Analysis
. The Fibonacci sequence is a sequence given by the equations:
F1 = 1, F2 = 1, Fn+2 = Fn+1 + Fn
So for example,
F3 = F2 + F1 =1+1=2
F1 = F3+ F2 = 2 +1 = 3
(a) (2 marks) Use the definition of the (n +1)t Fibonacci number to find a 2 × 2matrix A which satisfies the equation:
A\left(\begin{array}{c} F_{n} \\ F_{n+1} \end{array}\right)=\left(\begin{array}{c} F_{n+1} \\ F_{n+2} \end{array}\right)
(b) (8 marks) Compute the eigenvalues of the matrix A that you found in the previous question. For each eigenvalue, find a basis for the eigen space.
(c) (2 marks) Find an invertible matrix P and aA = PDP-1diagonal matrix D such that
(d) 4 marks Use the equation from part (a) to prove by induction on n that for alln > 0,
A^{n}\left(\begin{array}{l} F_{1} \\ F_{2} \end{array}\right)=\left(\begin{array}{l} F_{n+1} \\ F_{n+2} \end{array}\right)
(e) (4 marks) Use parts (c) and (d) to prove following formula for the nth Fibonacci number:
F_{n}=\frac{1}{\sqrt{5}}\left(\left(\frac{1+\sqrt{5}}{2}\right)^{n}-\left(\frac{1-\sqrt{5}}{2}\right)^{n}\right)
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