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

Q=x2+xy-3xz-y2/4+yz-9z2/4

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

    x2=x2+xy-3xz+y2/4+9z2/4-3yz/2

    Hence, we may re-express Q as

    Q=x2-yz/2
  • 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.

    Q=x2-y2-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

    y2=y2+yz/2+z2/16

    and hence

    Q=x2-y2+z2/16

    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