DLA Elementary nilpotent form

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Section 35.5 Elementary nilpotent form

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What.

$\begin{equation*} \inv{P} A P = \begin{bmatrix} 0 \\ 1 \amp 0 \\ \amp \ddots \amp \ddots \\ \amp \amp 1 \amp 0 \end{bmatrix}\text{.} \end{equation*}$

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When.

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Matrix

$A$ is nilpotent, with

$A^{n-1} \ne \zerovec\text{.}$

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How.

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Choose an index

$j$ such that the

$\nth[j]$ column of

$A^{n-1}$ is nonzero. Then take the columns of

$P$ to be

$\begin{align*} \uvec{p}_1 \amp = \uvec{e}_j, \amp \uvec{p}_2 \amp = A\uvec{e}_j, \amp \uvec{p}_3 \amp = A^2\uvec{e}_j, \amp \amp\dotsc, \amp \uvec{p}_n \amp = A^{n-1}\uvec{e}_j\text{,} \end{align*}$

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where

$\uvec{e}_j$ is the

$\nth[j]$ standard basis vector.

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Result.

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