Objective.
Orthogonally diagonalize the symmetric matrix
\begin{equation*}
A = \begin{pmatrix}
5 \amp -8 \amp 4 \\
-8 \amp 5 \amp -4 \\
4 \amp -4 \amp -1
\end{pmatrix}
\end{equation*}
Section 8.3 Orthogonally Diagonalization
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Find the eigenvalues and the corresponding eigenvectors.
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Let \(v_{1}=[1,-1,1/2], v_{2}=[1,0,-2]\) and \(v_{3}=[0,1,2]-\frac{[0,1,2]\cdot [1,0,-2]}{[1,0,-2]\cdot [1,0,-2]}[1,0,-2]=\left(\frac{4}{5},\,1,\,\frac{2}{5}\right)\)You must know why we are doing it in such way!
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Let \(u_{i}=\frac{u_{i}}{\|u_{i}\|}\text{.}\)
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Let\begin{equation*} P=[u_{1}\ u_{2} \ u_{3}]= \begin{pmatrix} \frac{2}{3} \amp -\frac{2}{3} \amp \frac{1}{3} \\ \frac{\sqrt{5}}{5} \amp 0 \amp -\frac{2\sqrt{5}}{5} \, \\ \frac{4\sqrt{5}}{15} \amp \frac{\sqrt{5}}{3} \amp \frac{2\sqrt{5}}{15} \end{pmatrix} \end{equation*}
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\(\displaystyle P^{-1}AP=P^{T}AP=\left(\begin{array}{rrr} 15 \amp 0 \amp 0 \\ 0 \amp -3 \amp 0 \\ 0 \amp 0 \amp -3 \end{array}\right)\)
There is a serious mistake. Please fix it!
Exercise: Orthogonally diagonalize the sym metric matrix
\begin{equation*}
A=\left(\begin{array}{rrrr}
1 \amp -2 \amp 0 \amp 0 \\
-2 \amp 1 \amp 0 \amp 0 \\
0 \amp 0 \amp 1 \amp -2 \\
0 \amp 0 \amp -2 \amp 1
\end{array}\right)
\end{equation*}
