diagonalization of quadratic form
A quadratic form may be diagonalized by the following procedure:
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1.
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.
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2.
By completing the square, define a new variable such that there are no cross-terms involving .
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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:
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Since appears, we do not need to perform a change of variables.
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We have the cross terms and . If we define , then
Hence, we may re-express as
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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 .
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Title | diagonalization of quadratic form |
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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 |