diagonalization of quadratic form
A quadratic form may be diagonalized by the following procedure:
Find a variable such that 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.
By completing the square, define a new variable such that there are no cross-terms involving .
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 appears, we do not need to perform a change of variables.
We have the cross terms and . If we define , then
Hence, we may re-express as
We must now repeat the procedure with the remaining variables, and . Since neither nor appears, we must make a change of variable. Let us define .
|Title||diagonalization of quadratic form|
|Date of creation||2013-03-22 14:49:34|
|Last modified on||2013-03-22 14:49:34|
|Last modified by||rspuzio (6075)|