The matrix $$A=\left[\begin{array}{ccc} 6 &3 &-3\cr -5 &-2 &5\cr -2 &-2 &5 \end{array}\right]$$ has eigenvalue $\lambda = 3$ repeated three times. It has an eigenspace of dimension 2 and one generalized eigenvector.

A. Find a basis for the $3$-eigenspace:

$\Biggl\lbrace$

$\left.\vphantom{\begin{array}{c}\!\strut\\\!\strut\\\!\strut\\\!\strut\\\end{array}}\right[$ $\left.\vphantom{\begin{array}{c}\!\strut\\\!\strut\\\!\strut\\\!\strut\\\end{array}}\right]$

,

$\left.\vphantom{\begin{array}{c}\!\strut\\\!\strut\\\!\strut\\\!\strut\\\end{array}}\right[$ $\left.\vphantom{\begin{array}{c}\!\strut\\\!\strut\\\!\strut\\\!\strut\\\end{array}}\right]$

$\Biggr\rbrace$

B. Find a generalized $3$-eigenvector, as well as the eigenvector it generalizes:

$\boldsymbol{\vec{w}} =$

$\left.\vphantom{\begin{array}{c}\!\strut\\\!\strut\\\!\strut\\\!\strut\\\end{array}}\right[$ $\left.\vphantom{\begin{array}{c}\!\strut\\\!\strut\\\!\strut\\\!\strut\\\end{array}}\right]$

generalizes the $3$-eigenvector $\boldsymbol{\vec{v}} =$

$\left.\vphantom{\begin{array}{c}\!\strut\\\!\strut\\\!\strut\\\!\strut\\\end{array}}\right[$ $\left.\vphantom{\begin{array}{c}\!\strut\\\!\strut\\\!\strut\\\!\strut\\\end{array}}\right]$