Linear transformations of a specific characteristic polynomial.

Question

I want to know all linear transformations having the characteristic polynomial $(x-1)^3(x+1)^2$?how can I know their number? how can I know them exactly? Is it by Jordan blocks or what?

Any help will be appreciated!

Answer 1

Since $(x-1)^3(x+1)^2$ splits over any field, a matrix with that characteristic polynomial has a Jordan Normal Form over the base field. The linear transformations are hence obtained by conjugating the basic JNF matrices \begin{align*} \begin{pmatrix} A & 0 \\ 0 & B \end{pmatrix} \end{align*} where \begin{align*} A \in \left\{ \begin{pmatrix} 1 & 1 & 0 \\ 0 & 1 & 1 \\ 0 & 0 & 1 \end{pmatrix}, \begin{pmatrix} 1 & 1 & 0 \\ 0 & 1 & 0 \\ 0 & 0 & 1 \end{pmatrix}, \begin{pmatrix} 1 & 0 & 0 \\ 0 & 1 & 0 \\ 0 & 0 & 1 \end{pmatrix}\right\} \end{align*} and

\begin{align*} B \in \left\{ \begin{pmatrix} -1 & 1 \\ 0 & -1 \end{pmatrix}, \begin{pmatrix} -1 & 0 \\ 0 & -1 \end{pmatrix}\right\} \end{align*}

If you're interested in the number of these over a finite field, say, you should probably ask a separate and more clear question about that.