(adsbygoogle = window.adsbygoogle || []).push({}); 1. The problem statement, all variables and given/known data

[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}.

3. 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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# Homework Help: Quadratic forms

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