latex

S_{N} = S_{N-2} + S_{N-1}

A = \left[\begin{array}{cc} 0 & 1\\ 1 & 1 \end{array}\right]

A \left[\begin{array}{cc} S_{N-2} \\S_{N-1} \end{array}\right] = 
\left[\begin{array}{cc} 0 & 1\\ 1 & 1 \end{array}\right] \left[\begin{array}{cc} S_{N-2} \\S_{N-1} \end{array}\right] = 
\left[\begin{array}{cc} S_{N-1} \\ S_{N} \end{array}\right]


\left[\begin{array}{cc} 0 & 1\\ 1 & 1 \end{array}\right] \left[\begin{array}{cc} 0 & 1\\ 1 & 1 \end{array}\right] \left[\begin{array}{cc} S_{0} \\S_{1} \end{array}\right] = 
\left[\begin{array}{cc} 0 & 1\\ 1 & 1 \end{array}\right] \left[\begin{array}{cc} S_{1} \\S_{2} \end{array}\right] = 
\left[\begin{array}{cc} 0 & 1\\ 1 & 1 \end{array}\right]^2 \left[\begin{array}{cc} S_{0} \\S_{1} \end{array}\right] = 
\left[\begin{array}{cc} S_{2} \\S_{3} \end{array}\right]

A = S \Lambda S^{-1}

A^2 = A = S \Lambda S^{-1} S \Lambda S^{-1} = S \Lambda I \Lambda S^T = S \Lambda^2 S^{-1}

A^N = S \Lambda^N S^{-1}

A^2 = A = S \Lambda S^{-1} S \Lambda S^{T} = S \Lambda I \Lambda S^T = S \Lambda^2 S^T




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