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Homediagonalization of quadratic form
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diagonalization of quadratic form
A quadratic form may be diagonalized by the following procedure:
1. Find a variable $x$ such that $x^{2}$ appears in the quadratic form. If no such variable can be found, perform a linear change of variable so as to create such a variable.
2. By completing the square, define a new variable $x^{{\prime}}$ such that there are no crossterms involving $x^{{\prime}}$.
3. Repeat the procedure with the remaining variables.
Example Suppose we have been asked to diagonalize the quadratic form
$Q=x^{2}+xy3xzy^{2}/4+yz9z^{2}/4$ 
in three variables. We could proceed as follows:

Since $x^{2}$ appears, we do not need to perform a change of variables.

We have the cross terms $xy$ and $3xz$. If we define $x^{{\prime}}=x+y/23z/2$, then
${x^{{\prime}}}^{2}=x^{2}+xy3xz+y^{2}/4+9z^{2}/43yz/2$ Hence, we may reexpress $Q$ as
$Q={x^{{\prime}}}^{2}yz/2$ 
We have a cross term $yz^{{\prime}}/2$. To eliminate this term, make a change of variable $y^{{\prime}}=y+z^{{\prime}}/4$. Then we have
${y^{{\prime}}}^{2}=y^{2}+yz^{{\prime}}/2+{z^{{\prime}}}^{2}/16$ and hence
$Q={x^{{\prime}}}^{2}{y^{{\prime}}}^{2}+{z^{{\prime}}}^{2}/16$ The quadratic form is now diagonal, so we are done. We see that the form has rank 3 and signature 2.
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