Fibonacci-numbers/Matrix/Iterations/Eigenvectors/Exercise

We consider the real matrix

${}M={\begin{pmatrix}1&1\\1&0\end{pmatrix}}\,.$

a) Determine

$M^{n}{\begin{pmatrix}1\\0\end{pmatrix}}$

for ${}n=1,2,3,4$.

b) Let

${}{\begin{pmatrix}x_{n+1}\\y_{n+1}\end{pmatrix}}:=M^{n}{\begin{pmatrix}1\\0\end{pmatrix}}\,.$

Establish a relation between the sequences ${}x_{n}$ and ${}y_{n}$, and determine a recursive formula for these sequences.

c) Determine the eigenvalues and the eigenvectors of ${}M$.

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