- #1
xman
- 92
- 0
i keep getting nonzero off diagonal elements when i try to reduce to simple sum of squares, of the equation
2 x_{1}^{2}+2x_{2}^{2}+x_{3}^{2}+2x_{1}x_{3}+2x_{2}x_{3}
what i have is
\left(\begin{array}{ccc} x_{1} & x_{2} & x_{3} \end{array}\right)<br /> \left(\begin{array}{ccc}<br /> 2 & 1 & 0 \cr<br /> 1 & 2 & 1 \cr<br /> 0 & 1 & 1 <br /> \end{array} \right) <br /> \left(\begin{array}{c} x_{1} \cr x_{2} \cr x_{3} \end{array} \right)<br />
so my thought was to calculate the eigenvalues of the coefficient matrix above, which yield complex solutions from the characteristic equation
1-6 \lambda+5 \lambda^{2}-\lambda^{3}=0
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
D = \left(\mid n \rangle \langle m \mid \right)^{T} A \left( \mid n \rangle \langle m \mid \right)
where the matrix
\left(\mid n \rangle \langle m \mid \right)
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.
2 x_{1}^{2}+2x_{2}^{2}+x_{3}^{2}+2x_{1}x_{3}+2x_{2}x_{3}
what i have is
\left(\begin{array}{ccc} x_{1} & x_{2} & x_{3} \end{array}\right)<br /> \left(\begin{array}{ccc}<br /> 2 & 1 & 0 \cr<br /> 1 & 2 & 1 \cr<br /> 0 & 1 & 1 <br /> \end{array} \right) <br /> \left(\begin{array}{c} x_{1} \cr x_{2} \cr x_{3} \end{array} \right)<br />
so my thought was to calculate the eigenvalues of the coefficient matrix above, which yield complex solutions from the characteristic equation
1-6 \lambda+5 \lambda^{2}-\lambda^{3}=0
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
D = \left(\mid n \rangle \langle m \mid \right)^{T} A \left( \mid n \rangle \langle m \mid \right)
where the matrix
\left(\mid n \rangle \langle m \mid \right)
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.
I was using my TI-89.