Section 35.5 Elementary nilpotent form
What.
\begin{equation*}
\inv{P} A P
= \begin{bmatrix} 0 \\ 1 \amp 0 \\ \amp \ddots \amp \ddots \\ \amp \amp 1 \amp 0 \end{bmatrix}\text{.}
\end{equation*}
When.
Matrix \(A\) is nilpotent, with \(A^{n-1} \ne \zerovec\text{.}\)
How.
Choose an index \(j\) such that the \(\nth[j]\) column of \(A^{n-1}\) is nonzero. Then take the columns of \(P\) to be
\begin{align*}
\uvec{p}_1 \amp = \uvec{e}_j, \amp
\uvec{p}_2 \amp = A\uvec{e}_j, \amp
\uvec{p}_3 \amp = A^2\uvec{e}_j, \amp
\amp\dotsc, \amp
\uvec{p}_n \amp = A^{n-1}\uvec{e}_j\text{,}
\end{align*}
where \(\uvec{e}_j\) is the \(\nth[j]\) standard basis vector.
Result.
Always exactly of the form described at the beginning of this section.