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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.

Result.

Always exactly of the form described at the beginning of this section.