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 cross-terms involving $x^{\prime }$.

3. Repeat the procedure with the remaining variables.

ExampleSuppose we have been asked to diagonalize the quadratic form

$$Q=x^2+xy-3xz-y^2/4+yz-9z^2/4$$

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 $x^{\prime }=x+y/2-3z/2$, then

  $$x^{\prime }{}^2=x^2+xy-3xz+y^2/4+9z^2/4-3yz/2$$

  Hence, we may re-express $Q$ as

  $$Q=x^{\prime }{}^2-yz/2$$

• We must now repeat the procedure with the remaining variables, $y$ and $z$. Since neither $y^2$ nor $z^2$ appears, we must make a change of variable. Let us define $z^{\prime }=z+2y$.

  $$Q=x^{\prime }{}^2-y^2-y{z}^{\prime }/2$$

• We have a cross term $-y{z}^{\prime }/2$. To eliminate this term, make a change of variable $y^{\prime }=y+z^{\prime }/4$. Then we have

  $$y^{\prime }{}^2=y^2+y{z}^{\prime }/2+z^{\prime }{}^2/16$$

  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.