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I am having a tough time understanding how to find the Jordan canonical form of a $5\times5$ matrix. This is a problem from my linear algebra book. Any help would be great or suggestions on how to get started.

Let $A$ be a $5\times5$ matrix with complex entries such that $A^{3} =0$. Find all the possible Jordan Canonical forms of $A$.

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    What does $A^3=0$ tell you about the eigenvalues?2012-07-23
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    Try to compute $A^3$ for different matrices in Jordan normal form, then you can get a feeling what $A^3=0$ means.2012-07-23
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    That the only eigenvalue is 0.2012-07-23
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    Thanks for the suggestions. I guess I don't really understand how you would compute $A^{3}$ for different matrices in Jordan normal form.2012-07-23
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    Now that you have the eigenvalues what can you say about the size of the Jordan blocks?2012-07-23
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    For example compute $\begin{pmatrix}0&1\\&0&1\\&&0&1\\&&&0&1\\&&&&0\end{pmatrix}^3$2012-07-23
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    Ok. I'll go out on a limb here, the size is 0? If it is, then I don't really know what it's supposed to look like. Maybe I am blocked tonight or something.2012-07-23
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    Hint: if you have an $k \times k$ Jordan block for eigenvalue $0$, it means there are $k$ vectors $v_1, \ldots, v_k$ such that $A v_1 = v_2,\ A v_2 = v_3, \ \ldots, A v_{k-1} = v_k, \ A v_k = 0$. So what is $A^3 v_1$?2012-07-23
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    You can see how power of Jordan form looks like at [Wikipedia](http://en.wikipedia.org/wiki/Jordan_normal_form#Powers).2012-07-23

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