The matrix $$A=\left[\begin{array}{ccc} -4 &-2 &2\cr 2 &1 &-3\cr 0 &1 &-3 \end{array}\right]$$ has eigenvalue $\lambda = -2$ repeated three times.

Find an $-2$-eigenvector for $\mathrm{A}$:

$\qquad \boldsymbol{\vec{v}} =$

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

Give a $\boldsymbol{\vec{v}}$-generalized $-2$-eigenvector:

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

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

Give a $\boldsymbol{\vec{w}}$-generalized $\boldsymbol{\vec{v}}$-generalized $-2$-eigenvector:

$\qquad \boldsymbol{\vec{u}} =$

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

(To be counted correct, all three vectors must be entered and be consistent.)