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I have a set of $A_{n\times n}$ matrices that satisfies $I+A+A^2+A^3+A^4 = 0$.

I can see that $A^5 = I \Rightarrow A^{k+5}=A^k$. How is this possible if A doesn't consists of a series of permutation matrices?

The homework question is: what could be said about $\dim(\operatorname{span}(I,A,A^2,A^3,....))$?

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    It does follow that $A$ must be similar to a permutation matrix. But that happens to have no bearing on the question you're asked. Note also that you are really only given one matrix which satisfies the equation given.2012-12-01
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    If $I + A + A^2 + A^3 + A^4 = 0$, then what can you say about the set $\{I, A, A^2, A^3, A^4\}$?2012-12-01
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    Thank you for your answers. I forgot to mention that I do realise the dimension must be at most 4. I'm trying to figure if more details could be inferred from the given equation.2012-12-01
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    If $A$ is a $2\times 2$ rotation matrix with angle $2\pi/5$ then $I+A+A^2+A^3+A^4=0$ (it's certainly true that $A^5=I$) but $A$ is not a permutation matrix (nor is it similar to one).2012-12-01
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    @user46450 Definitely a little more can be said if we know which field the entries of $A$ are taken from. For instance $\dim > 1$ if $A$ is real-valued (I think the dimension must be even, in fact).2012-12-01
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    Given the ldots in the problem statement, it is not unlikely that the problem poser merely expects "it is finite" as answer.2012-12-01
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    @HagenvonEitzen Given that it is a finite-dimensional vector space, I'm not sure that is quite enough. But I agree they are probably just looking for "$\le 4$", or at worst "$\le \min(4,n)$".2012-12-01
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    @ErickWong Oh,m yes, $n$ was supposed to be finite - my bad :)2012-12-01

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