- #1
xman
- 93
- 0
i keep getting nonzero off diagonal elements when i try to reduce to simple sum of squares, of the equation
[tex]2 x_{1}^{2}+2x_{2}^{2}+x_{3}^{2}+2x_{1}x_{3}+2x_{2}x_{3} [/tex]
what i have is
[tex] \left(\begin{array}{ccc} x_{1} & x_{2} & x_{3} \end{array}\right)
\left(\begin{array}{ccc}
2 & 1 & 0 \cr
1 & 2 & 1 \cr
0 & 1 & 1
\end{array} \right)
\left(\begin{array}{c} x_{1} \cr x_{2} \cr x_{3} \end{array} \right)
[/tex]
so my thought was to calculate the eigenvalues of the coefficient matrix above, which yield complex solutions from the characteristic equation
[tex] 1-6 \lambda+5 \lambda^{2}-\lambda^{3}=0 [/tex]
From the complex eigenvalues I obtain complex eigenvectors, which i'll post if necessary, but are rather lengthy. From the eigenvectors I choose to use Gram-Schmidt orthogonalization to form an orthonormal basis set. From which I construct a matrix with the corresponding basis set, and use diagonalize the system I have the diagonalization matrix
[tex] D = \left(\mid n \rangle \langle m \mid \right)^{T} A \left( \mid n \rangle \langle m \mid \right) [/tex]
where the matrix
[tex] \left(\mid n \rangle \langle m \mid \right) [/tex]
is the orthonormal eigenvector matrix. When I'm done with all of this I'm not getting a diagonalized matrix. I was wondering if I am making a mistake in my approach, or if anyone else does get a diagonalized matrix equation.
[tex]2 x_{1}^{2}+2x_{2}^{2}+x_{3}^{2}+2x_{1}x_{3}+2x_{2}x_{3} [/tex]
what i have is
[tex] \left(\begin{array}{ccc} x_{1} & x_{2} & x_{3} \end{array}\right)
\left(\begin{array}{ccc}
2 & 1 & 0 \cr
1 & 2 & 1 \cr
0 & 1 & 1
\end{array} \right)
\left(\begin{array}{c} x_{1} \cr x_{2} \cr x_{3} \end{array} \right)
[/tex]
so my thought was to calculate the eigenvalues of the coefficient matrix above, which yield complex solutions from the characteristic equation
[tex] 1-6 \lambda+5 \lambda^{2}-\lambda^{3}=0 [/tex]
From the complex eigenvalues I obtain complex eigenvectors, which i'll post if necessary, but are rather lengthy. From the eigenvectors I choose to use Gram-Schmidt orthogonalization to form an orthonormal basis set. From which I construct a matrix with the corresponding basis set, and use diagonalize the system I have the diagonalization matrix
[tex] D = \left(\mid n \rangle \langle m \mid \right)^{T} A \left( \mid n \rangle \langle m \mid \right) [/tex]
where the matrix
[tex] \left(\mid n \rangle \langle m \mid \right) [/tex]
is the orthonormal eigenvector matrix. When I'm done with all of this I'm not getting a diagonalized matrix. I was wondering if I am making a mistake in my approach, or if anyone else does get a diagonalized matrix equation.