Bilinear form/Symmetric/Type/Inertia law/Fact/Proof

Proof

With respect to an orthogonal basis ${}u_{1},\ldots ,u_{n}$ of ${}V$ (which exists due to fact), the Gram matrix is in diagonal form. Let ${}p'$ be the number of positive diagonal entries, and let ${}q'$ be the number of negative diagonal entries. We may order the basis in such a way that the first ${}p'$ diagonal entries are positive, the following ${}q'$ diagonal entries are negative, and the remaining entries are ${}0$. On the linear subspace ${}U=\langle u_{1},\ldots ,u_{p'}\rangle$ of dimension ${}p'$, the restricted bilinear form is positive definite; therefore, ${}p'\leq p$ holds. Let ${}W=\langle u_{p'+1},\ldots ,u_{n}\rangle$; on this linear subspace, the restricted bilinear form is negative semidefinite. We have ${}V=U\oplus W$, and these spaces are orthogonal to each other.

Assume now that there exists a linear subspace ${}U'$, such that the bilinear form restricted to is positive definite, and such that its dimension ${}p$ is larger than ${}p'$.

The dimension of ${}W$ is ${}n-p'$, therefore, ${}W\cap U'\neq 0$ because of fact. For a vector ${}w\in W\cap U'$, ${}w\neq 0$, we get directly the contradiction ${}\left\langle w,w\right\rangle >0$ and ${}\left\langle w,w\right\rangle \leq 0$.

To fact