Quadratic Mathematics | Part IB, 2002

Define the rank and signature of a symmetric bilinear form ϕ\phi on a finite-dimensional real vector space. (If your definitions involve a matrix representation of ϕ\phi, you should explain why they are independent of the choice of representing matrix.)

Let VV be the space of all n×nn \times n real matrices (where n2n \geqslant 2 ), and let ϕ\phi be the bilinear form on VV defined by

ϕ(A,B)=trABtrAtrB\phi(A, B)=\operatorname{tr} A B-\operatorname{tr} A \operatorname{tr} B

Find the rank and signature of ϕ\phi.

[Hint: You may find it helpful to consider the subspace of symmetric matrices having trace zero, and a suitable complement for this subspace.]

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