403/503 – HW6

February 24, 2015

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

Recall that the nth 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.