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Is there a 'quick way' of computing the rank and signature of the quadratic form $$q(x,y,z) = xy - xz$$ as I can only think of doing the huge computation where you find a basis such that the matrix of this quadratic form only has entries on the diagonal and compute it that way. Even doing that I must have made a mistake somewhere and seeing as this question is only worth 4 marks (it's a past exam question which doesn't have a mark scheme) then I thought that was rather a lot of work.

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It is enough to write the form as linear combination of squares (whose interior is linearly independent, though). Diagonalization is indeed overkill. –  Phira Apr 26 '12 at 13:33

2 Answers 2

up vote 3 down vote accepted

Hint:

1) $$ xy-xz=x(y-z) $$

2) $$ ab=\left(\frac{a+b}{2}\right)^2-\left(\frac{a-b}{2}\right)^2 $$

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A generalization, let $x,y,z$ be complex vectors of dimension $n$. Then the matrix $xy^*-xz^*=x(y-z)^*$ has rank at most one. The innertia contains $n-1$ zeros.

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