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Recurrence Relationsmedium
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The companion matrix form of the Fibonacci recurrence is (FnFn−1)=M(Fn−1Fn−2)\begin{pmatrix} F_n \\ F_{n-1} \end{pmatrix} = M \begin{pmatrix} F_{n-1} \\ F_{n-2} \end{pmatrix}(Fn​Fn−1​​)=M(Fn−1​Fn−2​​) where M=(1110)M = \begin{pmatrix} 1 & 1 \\ 1 & 0 \end{pmatrix}M=(11​10​). The characteristic roots of the Fibonacci recurrence Fn=Fn−1+Fn−2F_n = F_{n-1} + F_{n-2}Fn​=Fn−1​+Fn−2​ equal the eigenvalues of MMM. What are these eigenvalues?