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  "body": "\n## Summary\n\nA matrix difference equation describes how a vector of quantities changes in discrete time by multiplication by a matrix, possibly plus an input term.\nIt is the discrete analogue of a system of linear differential equations.\nEvery linear recurrence of higher order can be rewritten as a first-order matrix difference equation for a state vector.\nWith a constant matrix $A$, the homogeneous solutions are $x(n) = A^n x(0)$, so their behaviour is governed by the eigenvalues of $A$.\n\n## Explanations\n\n### L1 human\n\nThink of a state, such as the numbers of young and old animals in a population, recorded once a year.\nA matrix difference equation says that next year's state is a fixed matrix times this year's state.\nFor the Fibonacci numbers, the pair $(F_{n+1}, F_n)$ is obtained from $(F_n, F_{n-1})$ by the matrix with rows $(1, 1)$ and $(1, 0)$.\nSo repeated multiplication by one matrix generates the whole sequence.\n",
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