i keep getting nonzero off diagonal elements when i try to reduce to simple sum of squares, of the equation(adsbygoogle = window.adsbygoogle || []).push({});

[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.

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# Homework Help: Reduction of quadratic form (principal axis)

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