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I don't understand the 'completing the square' method for computing the rank and signature of a quadratic form. The rank and signature are defined to be $ p + q $ and $ p - q $ respectively, where $p$ and $q$ are the maximum dimensions of positive (resp. negative) definite subspaces of the space the quadratic form is defined on, but I just can't see how this method works.

Could anyone please explain?


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Try diagonalizing the form, which you can do when the form is non-degenerate. Then substract the negative eigenvalues from the positive eigenvalues. Sylvester's theorem guarantees the signature is independent of the choice of basis. –  gary May 31 '11 at 1:44
Sorry, I just realized you were referring to the method of 'completing the square', which I don't recognize by name, and not just finding the signature. Can you describe it for us? –  gary May 31 '11 at 1:46

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