Linear Algebra Done Right · Problem 5D.21

Inner Product Spaces — Sheldon Axler

Mathematics — Linear Algebra · diagonalizable-operators

Problem

The Fibonacci sequence F0,F1,F2,…F_0, F_1, F_2, \ldots is defined by F0=0,F1=1,andFn=Fn−2+Fn−1 for n≥2.F_0 = 0, \quad F_1 = 1, \quad \text{and} \quad F_n = F_{n-2} + F_{n-1} \text{ for } n \geq 2. Define T∈L(R2)T \in \mathcal{L}(\mathbf{R}^2) by T(x,y)=(y,x+y)T(x, y) = (y, x + y). ((a)) Show that Tn(0,1)=(Fn,Fn+1)T^n(0, 1) = (F_n, F_{n+1}) for each nonnegative integer nn. ((b)) Find the eigenvalues of TT. ((c)) Find a basis of R2\mathbf{R}^2 consisting of eigenvectors of TT. ((d)) Use the solution to (c) to compute Tn(0,1)T^n(0, 1). Conclude that Fn=15[(1+52)n−(1−52)n]F_n = \frac{1}{\sqrt{5}}\left[\left(\frac{1 + \sqrt{5}}{2}\right)^n - \left(\frac{1 - \sqrt{5}}{2}\right)^n\right] for each nonnegative integer nn. ((e)) Use (d) to conclude that if nn is a nonnegative integer, then the Fibonacci number FnF_n is the integer that is closest to 15(1+52)n.\frac{1}{\sqrt{5}}\left(\frac{1 + \sqrt{5}}{2}\right)^n.

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