Bilinear form/Symmetric/Type/Minor criterion/Fact

Minor criterion for the type

Let ${}\left\langle -,-\right\rangle$ be a symmetric bilinear form on a finite-dimensional real vector space ${}V$. Let ${}v_{1},\ldots ,v_{n}$ denote a basis of ${}V$. Let ${}G$ be the Gram matrix of ${}\left\langle -,-\right\rangle$ with respect to this basis. Suppose that the determinant ${}D_{k}$ of the square submatrices

${}M_{k}={\left(\left\langle v_{i},v_{j}\right\rangle \right)}_{1\leq i,j\leq k}\,$

are not ${}0$ for ${}k=1,\ldots ,n$. Let ${}a$ be the number of changes of sign in the sequence

$D_{0}=1,\,D_{1}=\det M_{1},\,D_{2}=\det M_{2},\ldots ,D_{n}=\det M_{n}=\det G.$

Then the

type

of ${}\left\langle -,-\right\rangle$ is ${}(n-a,a)$.

Proof, Write another proof