\[A\]
is symmetric, then the eigenvectors are orthogonal.Example:
\[A= \left( \begin{array}{ccc} 4 & 2 & 2 \\ 2 & 4 & 2 \\ 2 & 2 & 4 \end{array} \right) \]
.The eigenvalues of
\[A\]
are the solutions to \[det(A- \lambda I)=0\]
.\[\begin{equation} \begin{aligned} det(A- \lambda I) &=det(\left( \begin{array}{ccc} 4 & 2 & 2 \\ 2 & 4 & 2 \\ 2 & 2 & 4 \end{array} \right) - \lambda \left( \begin{array}{ccc} 1 & 0 & 0 \\ 0 & 1 & 0 \\ 0 & 0 & 1 \end{array} \right) ) \\ &= \left| \begin{array}{ccc} 4- \lambda & 2 & 2 \\ 2 & 4- \lambda & 2 \\ 2 & 2 & 4- \lambda \end{array} \right| \\ &= (4 -\lambda)((4- \lambda)^2-2^2)-2(2(4-\lambda) -2^2)+2(2(4-\lambda)-2^2) \\ &=(2- \lambda)^2(8- \lambda) = 0 \end{aligned} \end{equation}\]
.Hence
\[\lambda_1=2, \: \lambda_2 =8\]
.Now find the eigenvectors.
Let
\[\mathbf{v}=\begin{pmatrix}x_1\\x_2\\x_3\end{pmatrix}\]
\[\lambda_1=2\]
\[(A-2I) \mathbf{v}= \left( \begin{array}{ccc} 2 & 2 & 2 \\ 2 & 2 & 2 \\ 2 & 2 & 2 \end{array} \right)\begin{pmatrix}x_1\\x_2\\x_3\end{pmatrix}= \begin{pmatrix}2x_1+2x_2+2x_3\\2x_1+2x_2+2x_3\\2x_1+2x_2+2x_3\end{pmatrix} = \begin{pmatrix}0\\0\\0\end{pmatrix}\]
Set
\[x_2=1, \: x_3=0\]
then \[x_1=-1\]
. An eigenvector is \[\begin{pmatrix}-1\\1\\0\end{pmatrix}\]
Set
\[x_2=0, \: x_3=1\]
then \[x_1=-1\]
. An eigenvector is \[\begin{pmatrix}-1\\0\\1\end{pmatrix}\]
\[\lambda_2=8\]
\[(A-2I) \mathbf{v}= \left( \begin{array}{ccc} -4 & 2 & 2 \\ 2 & -4 & 2 \\ 2 & 2 & -4 \end{array} \right)\begin{pmatrix}x_1\\x_2\\x_3\end{pmatrix}= \begin{pmatrix}-4x_1+2x_2+2x_3\\2x_1-4x_2+2x_3\\2x_1+2x_2-4x_3\end{pmatrix} = \begin{pmatrix}0\\0\\0\end{pmatrix}\]
A solution is
\[x_1=x_2=x_3=1\]
then the last eigenvector is \[\begin{pmatrix}1\\1\\1\end{pmatrix}\]
The orthogonal eigenvectors are
\[\left\{ \begin{pmatrix}-1\\0\\1\end{pmatrix} , \begin{pmatrix}-1\\1\\0\end{pmatrix} , \begin{pmatrix}1\\1\\1\end{pmatrix} \right\}\]
.We can find an orthonormal set of vectors by normalising each eigenvector. We get
\[\left\{ \begin{pmatrix}{-1/ \sqrt{2}}\\0\\{1\sqrt{2}}\end{pmatrix} , \begin{pmatrix}{-1\sqrt{2}}\\{1\sqrt{2}}\\0\end{pmatrix} , \begin{pmatrix}{1/\sqrt{3}}\\{1/\sqrt{3}}\\{1/\sqrt{3}}\end{pmatrix}\right\} \]
.