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.
$\endgroup$ 31 Answer
$\begingroup$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.
