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I asked this question the other day but I still didn't understand it. Hopefully someone can get through to me this time.

I have a question that reads:

Diagonalize the quadratic form A(x,y) = 3x^2 - 12xy + 7y^2.

I am unsure what diagonalization is in this context. I know what it means to diagonalize a matrix, say, A. You find it's eigenvectors, put them into a matrix, say S, find that matrix's inverse, and work out S^-1AS.

If someone could explain, that would be much appreciated.

Thanks!

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Diagonalize a quadratic form means diagonalize the matrix of this quadratic form which is also the matrix of the polar form associated to it. In your example this matrix is

$$M=\begin{pmatrix}3&-6\\-6&7\end{pmatrix}$$ and notice that

$$A(x,y)=X^TMX$$ where $X=(x,y)^T$

Added$\quad$ There's a method due to Gauss and it consists to complete the square: so we get

$$A(x,y)=3(x-2y)^2-5y^2$$ so with the new variables $$x_1=(x-2y)\quad; \quad y_1= y$$ we get

$$A(x,y)=X_1^T\operatorname{diag}(3,-5)X_1=(PX)^T\operatorname{diag}(3,-5)PX$$

where is the transition matrix

$$P=\begin{pmatrix}1&-2\\0&1\end{pmatrix}$$

Remark$\quad$ If we want use the first method i.e. diagonalize the matrix then we should do it via an orthonormal basis and this is possible since the matrix $M$ is symmetric real.

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  • $\begingroup$ You, sir, are a genie. I know I'm not supposed to write thank-yous here, but I'm going to do it anyway. Thanks! :D $\endgroup$ – Arbitrationer Nov 9 '14 at 18:41
  • $\begingroup$ You're welcome:-) $\endgroup$ – user63181 Nov 9 '14 at 18:42

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