Hermitian form/Type via self-adjoint endomorphism/Fact

Eigenvalue criterion for the type (Hermitian form)

Let ${}V$ be a finite-dimensional ${}{\mathbb {K} }$-vector space, endowed with an inner product. Let ${}\Psi$ denote an Hermitian form on ${}V$, corresponding to the self-adjoint endomorphism

$\varphi \colon V\longrightarrow V$

in the sense of fact. Let ${}(p,q)$ be the type of ${}\Psi$.

Then ${}p$ is the number of

positive eigenvalues, and ${}q$ is the number of negative eigenvalues of ${}\varphi$, where we have to take this numbers with their (algebraic or geometric)

multiplicity.

Proof, Write another proof