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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.