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Google has been surprisingly unhelpful for me.

A homework problem from my algebra class asks me to

Calculate p(A) where A is a Jordan cell and p is a polynomial.

I'd like to attempt the problem on my own, but I admit that without an inkling of what a "Jordan cell" is, this task seems rather daunting.

A simple definition an undergraduate student taking a course in algebra can understand would be much appreciated.

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Try this query: google.com/search?q=Jordan+cell+polynomial –  wj32 Nov 24 '12 at 9:13
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@math4tots: I guess that the usual notation is Jordan block. Anyway, check: en.wikipedia.org/wiki/Jordan_matrix –  Dennis Gulko Nov 24 '12 at 9:15
    
We have used the term Jordan cell and it's indeed the same as a Jordan block. –  nonpop Nov 24 '12 at 9:45

1 Answer 1

up vote 1 down vote accepted

To calculate $p(A)$, first calculate $A^k$ for all $k$. For $$A=\begin{pmatrix}\lambda &1&0&\cdots &0\\ 0&\lambda&1&\cdots&0\\ \vdots&&\ddots&&\vdots\\0&0&0&&1\\0&0&0&\cdots&\lambda\end{pmatrix}$$ You can prove by induction that $A^k=(b_{i,j}^k)_{i,j}$ where $b_{i,j}^k=\binom{k}{j-i}\lambda^{k-(j-i)}$. (remember that $\binom{k}{j-i}=0$ when $j-i<0$ or $j-i>k$)
From here finding $p(A)=a_kA^k+...+a_1A+a_0I$ is easy.

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