Back when I was learning Linear Algebra, I always found it ridiculously cool that one could reduce a wide range of recursive sequences to discrete functions by representing them as repeated applications of a matrix. The matrix could then be diagonalized and an analytic solution found. One cool example was finding the discrete function that describes the Fibonacci sequence. I decided to explore it a little deeper and share it with you. It turns out that the Fibonacci sequence is intimately related to the Golden Ratio; in fact, it's one of its eigenvalues! Moreover, there are continuous and complex extensions to the sequence.
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