$ A=\mat{ 0&-1&2\\ -1&0&-2\\ 2&-2&3 } $ $A$ is a [[../02 Areas/Math/Symmetric Matrix|Symmetric Matrix]] $ \begin{align} \det(A-tI) &= \mat{ -t & -1 & 2 \\ -1 & -t & -2 \\ 2 & -2&3-t } \\ &= -t^3 + 3t^2 + 9t + 5 \end{align} $ $\begin{align} t+1\overline{) t^3 + 3t^2 + 9t +5} &= (-t^2 + 4t+5)(t+1) \\ -t^2 + 4t+5 &= 0 \\ t&= \set{ -1, -5 }\\ 0&= (t-5)(t+1)(t+1)\\ \end{align} $ $ \begin{align} &\lambda_{1}=5& &\lambda_{2}=1 & &\lambda_{3} = 1 \end{align} $ $ \begin{align} \dim E_{5} &= 1\\ \dim E_{-1} &= 2 \end{align} $ $\begin{align} E_{5} &= \op{Nul}\mat{ -5&-1&2\\ -1&-5&-2 \\2&-2&-2 }\\ &\sim \augmented{ccc|c}{ -5&-1&2&0\\ -1&-5&-2&0 \\2&-2&-2&0 }\\ &\sim \begin{cases} x_{1}&= \frac{1}{2} t \\ x_{2}&= -\frac{1}{2}t \\ x_{3} &= t \end{cases}\\ \vec x &= t\mat{\frac{1}{2}\\-\frac{1}{2}\\1} \end{align} $ $ \begin{align} E_{-1} &= \op{Nul}\mat{ 1&-1&2\\ -1&1&-2\\ 2&-2&4 }\\ &\sim \augmented{ccc|c}{ 1&-1&2&0\\ -1&1&-2&0\\ 2&-2&4&0 } \\&\sim \augmented{ccc|c}{ 1&-1&2&0\\ 0&0&0&0\\ 0&0&0&0 }\\ &\sim \begin{cases} x_{1}= t-2s\\ x_{2}=t\\ x_{3}=s \end{cases}\\ \vec x &= t\mat{1\\1\\0} + s\mat{-2\\0\\1} \end{align} $ $\begin{align} \vec u_{1} &= \mat{\frac{1}{2}\\-\frac{1}{2}\\1} \\ \vec u_{2} &= \mat{1\\1\\0} \\ \vec u_{3} &= \mat{-2\\0\\1} \end{align} $ $ \begin{align} \vec v_{2} &= \vec u_{2} = \mat{1\\1\\0}\\ \vec v_{3} &= \vec u_{3}- \frac{{\vec v_{2}^\intercal \vec u }}{\vec v_{2}^\intercal \vec v_{2}} \vec v_{2} =\mat{ -1\\1\\1 }\\ \mathcal B_{E_{-1}} &= \left\{ \mat{1\\1\\0},\mat{-1\\1\\1} \right\}\\ \end{align} $ $ \begin{align} \end{align} $