Quadratic Mathematics | Part IB, 2001

Let ϕ\phi be a symmetric bilinear form on a finite dimensional vector space VV over a field kk of characteristic 2\neq 2. Prove that the form ϕ\phi may be diagonalized, and interpret the rank rr of ϕ\phi in terms of the resulting diagonal form.

For ϕ\phi a symmetric bilinear form on a real vector space VV of finite dimension nn, define the signature σ\sigma of ϕ\phi, proving that it is well-defined. A subspace UU of VV is called null if ϕU0\left.\phi\right|_{U} \equiv 0; show that VV has a null subspace of dimension n12(r+σ)n-\frac{1}{2}(r+|\sigma|), but no null subspace of higher dimension.

Consider now the quadratic form qq on R5\mathbb{R}^{5} given by

2(x1x2+x2x3+x3x4+x4x5+x5x1)2\left(x_{1} x_{2}+x_{2} x_{3}+x_{3} x_{4}+x_{4} x_{5}+x_{5} x_{1}\right)

Write down the matrix AA for the corresponding symmetric bilinear form, and calculate detA\operatorname{det} A. Hence, or otherwise, find the rank and signature of qq.

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