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T-7
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Homework Statement
[tex]x^{2} + 2y^{2} + z^{2} + 2xy + 4xz + 6yz[/tex]
Write down the symmetric matrix A for which the form is expressible as [tex]x^{t}Ax[/tex] where t denotes transpose. Diagonalise each of the forms and in each case find a real non-singular matrix P for which the matrix [tex]P^{t}AP[/tex] is diagonal with entries in {1,-1,0}.
The Attempt at a Solution
I first tried this by completing the square.
[tex]
x^{2} + 2y^{2} + z^{2} + 2xy + 4xz + 6yz
= (x + y + 2z)^{2} + y^{2} - 3z^{2} + 2yz
= (x + y + 2z)^{2} + (y + z)^2 - 4z^2
= x_{1}^{2} + x_{2}^{2} - x_{3}^{2}[/tex]
where
[tex]
x_{1} = x + y + 2z,
x_{2} = y + z,
x_{3} = 2z,
[/tex]
However, I just can't seem to find the eigenvalues for this form.
The symmetric matrix A for this quadratic form is
[tex]
\[ \left( \begin{array}{ccc}
1 & 1 & 2 \\
1 & 2 & 3 \\
2 & 3 & 1 \end{array} \right)\]
[/tex]
and the characteristic polynomial is given by
[tex]
\[ \chi(\lambda) = \left| \begin{array}{ccc}
1-\lambda & 1 & 2 \\
1 & 2-\lambda & 3\\
2 & 3 & 1-\lambda \end{array} \right|.\]
[/tex]
I find this comes to
[tex]f(\lambda) = \lambda^{3} - 4(\lambda^2) + 9(\lambda) - 4[/tex]
which does not factorise -- so I can't get the eigenvalues, and can't form a matrix P. However, I have shown that a diagonal form is possible by completing the square. So surely I ought to be able to find three eigenvalues? Can someone point out where I've gone wrong?
Cheers!
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