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 $x'$ such that there are no cross-terms involving $x'$ .
3. Repeat the procedure with the remaining variables.

Example Suppose we have been asked to diagonalize the quadratic form $$Q = x^2 + xy - 3xz - y^2/4 + yz - 9 z^2/4$$ 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 $-3 xz$ . If we define $x' = x + y/2 - 3z/2$ , then $${x'}^2 = x^2 + xy - 3xz + y^2/4 + 9 z^2/4 - 3yz/2$$ Hence, we may re-express $Q$ as $$Q = {x'}^2 - yz/2$$
• 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 $z' = z + 2y$ . $$Q = {x'}^2 - y^2 - yz'/2$$
• We have a cross term $- yz'/2$ . To eliminate this term, make a change of variable $y' = y + z'/4$ . Then we have $${y'}^2 = y^2 + yz'/2 + {z'}^2/16$$ and hence $$Q = {x'}^2 - {y'}^2 + {z'}^2/16$$ The quadratic form is now diagonal, so we are done. We see that the form has rank 3 and signature 2.