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What sort of criterion is there for determining whether a matrix is nilpotent?

Specifically, I'm interested in the nilpotent matrices over finite fields. I realize that any such matrix will have to be singular, but if one repeatedly exponentiates a singular matrix over a finite field it will either return to its original value (without turning into the identity first) or turn into the zero matrix. So how can one know how the matrix is going to behave?

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    In other words, the fact that the entries are from a finite field doesn't really help you answering this question at all. Except possibly in the sense that you will know in advance that the 'size' of the entries in the powers of your matrix will remain below a certain bound (if you write a program to do this test for ya).2011-10-09

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If $A$ is $n\times n$ and nilpotent, then its characteristic polynomial will be $t^n$ (or $(-1)^nt^n$, depending on how exactly you define the characteristic polynomial).

By the Cayley-Hamilton Theorem, evaluating the polynomial at $A$ will be zero; thus, if $A$ is nilpotent, then necessarily $A^n = 0$. Conversely, if $A^n=0$, then $A$ is nilpotent.

Thus, it suffices to check if $A^n=0$. You can do this in $\lceil\log_2(n)\rceil$ steps by repeated squaring.

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    And the downvote was because...?2011-10-05