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- #1

- Feb 5, 2012

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Here's a question with the summary of my method of how to solve it. I would really appreciate if you could go through it and let me know if there are any mistakes with my approach. Also are there any easier methods?

**Problem:**

Find an orthogonal transformation that reduces the following quadratic form to the principal axes.

\[q(x_1,\,x_2,\,x_3)=6x_{1}^{2}+5x_{2}^{2}+7x_{3}^{2}-4x_1 x_2+4x_1 x_3\]

**Solution:**

We reduce this to get rid of the cross terms as explained >>here<<.

\[q(x_1,\,x_2,\,x_3)=3\left(\frac{2x_1+2x_2-x_3}{3}\right)^2+9\left(\frac{-2x_1+x_2-2x_3}{3}\right)^2+6\left(\frac{-x_1+2x_2+2x_3}{3}\right)^2\]

So the matrix of the orthogonal transformation will be,

\[\begin{pmatrix}\frac{2}{3}&\frac{2}{3}&-\frac{1}{3}\\-\frac{2}{3}&\frac{1}{3}&-\frac{2}{3}\\-\frac{1}{3}&\frac{2}{3}&\frac{2}{3}\end{pmatrix}\]

Am I correct?