diagonalization of quadratic form

A quadratic formMathworldPlanetmath may be diagonalized by the following procedure:

  1. 1.

    Find a variable x such that x2 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. 2.

    By completing the square, define a new variable x such that there are no cross-terms involving x.

  3. 3.

    Repeat the procedure with the remaining variables.

Example Suppose we have been asked to diagonalize the quadratic form


in three variables. We could proceed as follows:

  • Since x2 appears, we do not need to perform a change of variables.

  • We have the cross terms xy and -3xz. If we define x=x+y/2-3z/2, then


    Hence, we may re-express Q as

  • We must now repeat the procedure with the remaining variables, y and z. Since neither y2 nor z2 appears, we must make a change of variable. Let us define z=z+2y.

  • We have a cross term -yz/2. To eliminate this term, make a change of variable y=y+z/4. Then we have


    and hence


    The quadratic form is now diagonal, so we are done. We see that the form has rank 3 and signaturePlanetmathPlanetmath 2.

Title diagonalization of quadratic form
Canonical name DiagonalizationOfQuadraticForm
Date of creation 2013-03-22 14:49:34
Last modified on 2013-03-22 14:49:34
Owner rspuzio (6075)
Last modified by rspuzio (6075)
Numerical id 7
Author rspuzio (6075)
Entry type Algorithm
Classification msc 15A03
Related topic DiagonalQuadraticForm