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So there is a formula for the $n$th power of a matrix in Jordan normal form. Is there a formula for the $n$th power of a general triangular matrix? If not, are there known formulas for "nice" upper triangular matrices? Like those consisting of all 1s and 0s.

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    For an upper triangular matrix that can be written as $D + N$ where $D$ is diagonal and $N$ is strictly upper-triangular (hence nilpotent) _and such that $D$ and $N$ commute_ (in particular if $D$ is scalar), it is very easy to compute the powers explicitly in terms of the powers of $D$ and $N$. In general I think you should just find the Jordan normal form or use special structure of the matrix in question.2011-10-05
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    Well, the diagonal elements are easy, and there's a neat formula for the superdiagonal entries, involving sums of the form $\sum\limits_{k=0}^{n-1} x^k y^{n-k-1}$...2011-10-05

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