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When a matrix $A$ is diagonalizble, i.e. $A = P \Lambda P^{-1}$, then it is very easy to compute $A^i$: it is just $P \Lambda^i P^{-1}$ where $\Lambda^i$ is the diagonal matrices with the eigenvalues exponentiated.

I have an upper triangular matrix where the diagonal is all 0s (this means that all the eigenvalues are 0). I would expect somehow to be able to do something similar to the exponentiation of $A$ above, but that's not possible because such upper triangular matrix is not diagonalizible. Is there another efficient way of computing the power of an upper triangular matrix with all 0s on the diagonal?

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You don't need to worry too much since the matrix is nilpotent. There are powers less than the size of the matrix to be computed and you might hardly ever need a closed form for powers. – user13838 Oct 13 '11 at 18:09
This is not exactly the same question, but it comes close:… – joriki Oct 13 '11 at 19:11
unfortunately @percusses remark should be extended insofar that this does not extend to negative and/or fractional powers which is otherwise an "automatic" feature as long as the entries of $\small \Lambda $ are nonzero or even positive and one can get pretty used to (and then careless with) the possibility of fractional powers with the help of diagonalization... – Gottfried Helms Oct 13 '11 at 19:41

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