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Math Help - Canonical form of a quadratic form

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    Canonical form of a quadratic form

    Write the quadratic form x_1x_2 +x_2x_3 +x_3x_1 in canonical form over \mathbb{C}.


    I started off by finding a symmetric matrix A for the quadratic form Q(\vec{x},\vec{x})=\vec{x}^TA\vec{x}, which is
    \begin{pmatrix} 0 & \frac{1}{2} & \frac{1}{2}\\ \frac{1}{2} & 0 & \frac{1}{2} \\ \frac{1}{2} & \frac{1}{2} & 0\end{pmatrix} with eigenvalues of \lambda=-.5, -.5, 1 and eigenvectors (-1,1,0), (-1,0,1), (1, 1, 1) respectively.

    I don't see how it's now possible to rewrite the expression as Q(\vec{x},\vec{x})=\lambda_1x_1^2+\lambda_2x_2^2+\  lambda_3x_3^2. And where do complex numbers have anything to do with the problem?
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    MHF Contributor FernandoRevilla's Avatar
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    Quote Originally Posted by davesface View Post
    ... with eigenvalues of \lambda=-.5, -.5, 1 and eigenvectors (-1,1,0), (-1,0,1), (1, 1, 1) respectively.
    Right.

    I don't see how it's now possible to rewrite the expression as Q(\vec{x},\vec{x})=\lambda_1x_1^2+\lambda_2x_2^2+\  lambda_3x_3^2.
    You need an orthormal basis of eigenvectors B=\{e_1,e_2,e_3\} (with the usual inner product). If P=[e_1\;e_2\;e_3] then,

    P^tAP=\textrm{diag}(-1/2,-1/2,1) .

    And where do complex numbers have anything to do with the problem?
    Absolutely nothing. Any cuadratic form on \mathbb{R} is diagonalizable on \mathbb{R} .

    Regards.

    Fernando Revilla
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