For each of the following, factor the matrix $A$ into a product $QDQ^{T}$ where $Q$ is orthogonal and $D$ is diagonal.

(a) $A = \left[\begin{array}{ccc} -5 &1 &1\cr 1 &4 &-8\cr 1 &-8 &4 \end{array}\right]$

$Q=$

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

$\qquad D=$

$\left.\vphantom{\begin{array}{c}\!\strut\\\!\strut\\\!\strut\\\!\strut\\\end{array}}\right[$ 0 0 $\left.\vphantom{\begin{array}{c}\!\strut\\\!\strut\\\!\strut\\\!\strut\\\end{array}}\right]$ 0 0 0 0

(b) $A = \left[\begin{array}{ccc} 0 &-2 &2\cr -2 &-1 &1\cr 2 &1 &-1 \end{array}\right]$

$Q=$

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

$\qquad D=$

$\left.\vphantom{\begin{array}{c}\!\strut\\\!\strut\\\!\strut\\\!\strut\\\end{array}}\right[$ 0 0 $\left.\vphantom{\begin{array}{c}\!\strut\\\!\strut\\\!\strut\\\!\strut\\\end{array}}\right]$ 0 0 0 0