1.I.8F

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

Let $V$ be the space of all $n \times n$ real matrices (where $n \geqslant 2$ ), and let $\phi$ be the bilinear form on $V$ defined by

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

Find the rank and signature of $\phi$.

[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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