| course |
course_year |
question_number |
tags |
title |
year |
Quadratic Mathematics |
IB |
62 |
IB |
2002 |
Quadratic Mathematics |
|
1.I.8F |
2002 |
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.]