Let $g$ be a bilinear form over $\mathbb{R}^8$ with signature $(5,3)$.
True or false - There exists a unique five dimensional subspace such that $g$ is positive-definite on this subspace.
My intuition says it is true, and that the subspace is a span of five vectors which make the five $+1$ in the representation matrix, but after proving existence, I am stuck with uniqueness. So far what I have concluded:
There exists a basis $v_1,\ldots,v_8$ such that $g(v_i,v_j) = \delta_{i,j},\ i,j\leq5$, $g(v_i,v_j) = -\delta_{i,j},\ 6\leq i,j\leq 8$. Suppose there is another 5 dimensional subspace such that $g$ is positive-definite on this subspace. We know for sure that $v_6,v_7,v_8$ are not in this subspace.
I have tried to show that a vector in this subspace is in the span I mentioned above, but no success.
Any suggestions? Is the statement even true?
Thanks
The restriction of the form to the subspace $\langle e_1, \ldots, e_5\rangle$ is positive, indeed the matrix of the form is $I_5$. Wiggle a bit the the $e_i$, $i=1,5$ to $e'_i$. The matrix of the form $( (e'_i, e'_j))_{1 \le i,j \le 5}$ changes a bit so for small variations it remains positive definite. Hence for $e'_i$ close to $e_i$ the restriction to $\langle e'_1, \ldots e'_5\rangle$ is still positive definite ( one can provide some effective estimates).