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I am trying to solve the question about jordan canonical form.

Let $F$ be a field and let $J$ be the matrix of $M_n(F)$ all of entries are $1$. Find the jordan canonical form.

So I barely know things about jordan canonical forms. I can compute them if minimal polynomial given other than that my knowledge is limited. Can anybody help me solve this question? How to start and what to do next?

Thanks everyone.

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  • $\begingroup$ Did you learn about eigenvalues and eigenvectors? $\endgroup$ – Mark Mar 26 '14 at 5:01
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If $e=(1,1,...,1)^T$, then $J=e e^T$. From this you can figure out all eigenvalues and relevant eigenspaces, from which the Jordan form follows.

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The span of the column vectors of the matrix is a line, and so the rank is 1. Thus, all the eigenvalues are $0$ except one of them which you can determine is $n$. Since you can find an $n-1$-dimensional $0$-eigenspace, there can be no nilpotency. This allows you to colnclude the Jordan form of $J$.

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