Hey, the task is as follows. Given [MATH]R=\frac{1}{7}
\begin{pmatrix}
-6 & 3 & 2\\
3 & 2 & 6\\
2 & 6 &-3
\end{pmatrix}
[/MATH]Find an orthogonal matrix S that diagonalizes R. What I did was first look for the Eigenvalues, [MATH]Det(\lambda I-R)=0[/MATH] and this will eventually yield the eigenvalues {-1,-1,1}. I then extract my eigenvectors: {-2,0,1},{-3,1,0},{1,3,2} and write them as a matrix:
[MATH]S= \begin{pmatrix} -2 & -3 & 1\\ 0 & 1 & 3\\ 1 & 0 &-2 \end{pmatrix} [/MATH]and the matrix D:
[MATH]D= \begin{pmatrix} 1 & 0 & 0\\ 0 & -1 & 0\\ 0 & 0 &-1 \end{pmatrix} [/MATH]How do I know then find an orthogonal matrix S that diagonalizes R? What I know is that [MATH]D=S^{-1}AS[/MATH] where D is the matrix containing the eigenvalues and S is the matrix containing the eigenvectors. From my understanding I have to make is so that [MATH]D=S^{-1}AS=S^TAS[/MATH] but it is not immediately clear to me how I am supposed to achieve that.
[MATH]S= \begin{pmatrix} -2 & -3 & 1\\ 0 & 1 & 3\\ 1 & 0 &-2 \end{pmatrix} [/MATH]and the matrix D:
[MATH]D= \begin{pmatrix} 1 & 0 & 0\\ 0 & -1 & 0\\ 0 & 0 &-1 \end{pmatrix} [/MATH]How do I know then find an orthogonal matrix S that diagonalizes R? What I know is that [MATH]D=S^{-1}AS[/MATH] where D is the matrix containing the eigenvalues and S is the matrix containing the eigenvectors. From my understanding I have to make is so that [MATH]D=S^{-1}AS=S^TAS[/MATH] but it is not immediately clear to me how I am supposed to achieve that.
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