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Get the following quadratic form:

$$Q(x)=x_1^{2}+x_3^{2}+4x_1x_2−4x_1x_3$$ to the canonical form using orthogonal transformations.

I have no idea how to solve that exercise.

Please,could you help me at this one?

I thank you for your understanding and I wait forward your response!

Adrian Berteanu
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    write the Hessian matrix of second partial derivatives, or half that since it will still be integers. Find the eigenvalues and make a matrix $P$ with columns the eigenvectors normalized to length $1.$ Then $P^T H P$ is diagonal and $P^T P = I$ – Will Jagy Jan 03 '16 at 19:15
  • The bad news is that the characteristic polynomial of $H,$ which I called $M$ in my previous answer http://math.stackexchange.com/questions/1597332/a-question-about-a-canonical-form-of-a-quadratic-form-using-gauss-theorem/1597440#1597440 has three distinct irrational real roots, meaning we are dealing with https://en.wikipedia.org/wiki/Casus_irreducibilis and nothing can be done by hand calculations. In some courses they would want numerical calculations. What is the source of this problem? – Will Jagy Jan 03 '16 at 19:37
  • This problem was given to us by a student in the senior years who told us that this was the subject of the previous year exam at geometry. – Adrian Berteanu Jan 03 '16 at 19:39

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