## 403/503 – HW6

This exercise is due Tuesday, March 3, at the beginning of lecture.

Recall that the $n$th Jordan block for $\lambda$, $J(\lambda,n)$, is the $n\times n$ matrix whose entries along the main diagonal are $\lambda$, along the diagonal immediately below the main one are $1$, and all other entries are $0$. For example, $J(5,4)$ is the matrix

$\displaystyle \left(\begin{array}{cccc}5&0&0&0\\ 1&5&0&0\\ 0&1&5&0\\ 0&0&1&5\end{array}\right).$

Find a general formula for the powers of Jordan blocks, i.e., compute $J(\lambda,n)^k$.