What is an example of a quadratic form $Q$ on real vector space $V$ such that the maximal positive definite subspace and the maximal negative definite subspace are not uniquely determined?
By Sylvester's law of inertia, the dimensions of these subspaces are uniquely determined. I think (?) this means they both have to have dimension at least 3.
Should we do this by completing the square in two different ways (e.g. $x_1^2 + x_2^2 + x_3^2 - x_4^2 - x_5^2 - x_6^2 = (x_1 + x_2 + x_3)^2 - (x_1 + x_2)^2 + ... - ...$ ), or by considering bases and subspaces explicitly?
Many thanks for any help with this!