Hermitian form/Type via self-adjoint endomorphism/Fact/Proof

Proof

According to fact, the characteristic polynomial of ${}\varphi$ splits into real linear factors. Let ${}\lambda _{1},\ldots ,\lambda _{r}$ be the positive zeroes and let ${}\lambda _{r+1},\ldots ,\lambda _{s}$ be the negative zeroes. Due to fact, we have a direct sum decomposition

$V=\operatorname {Eig} _{\lambda _{1}}{\left(\varphi \right)}\oplus \cdots \oplus \operatorname {Eig} _{\lambda _{r}}{\left(\varphi \right)}\oplus \operatorname {Eig} _{\lambda _{r+1}}{\left(\varphi \right)}\oplus \cdots \oplus \operatorname {Eig} _{\lambda _{s}}{\left(\varphi \right)}\oplus \operatorname {Eig} _{0}{\left(\varphi \right)}\,,$

which is orthogonal with respect to the inner product (${}\operatorname {Eig} _{0}{\left(\varphi \right)}$ might be zero). For vectors ${}v_{i}$ and ${}v_{j}$ from different eigenspaces, we have

${}\Psi (v_{i},v_{j})=\left\langle \varphi (v_{i}),v_{j}\right\rangle =\left\langle \lambda _{i}v_{i},v_{j}\right\rangle =\lambda _{i}\left\langle v_{i},v_{j}\right\rangle =0\,;$

therefore, the eigenspaces are also orthogonal with respect to the form ${}\Psi$. For

${}v=v_{1}+\cdots +v_{r}\neq 0\,$

with ${}v_{i}\in \operatorname {Eig} _{\lambda _{i}}{\left(\varphi \right)}$, we have

${}{\begin{aligned}\Psi (v,v)&=\Psi (v_{1},v_{1})+\cdots +\Psi (v_{r},v_{r})\\&=\Psi _{\varphi }(v_{1},v_{1})+\cdots +\Psi _{\varphi }(v_{r},v_{r})\\&=\left\langle \varphi (v_{1}),v_{1}\right\rangle +\cdots +\left\langle \varphi (v_{r}),v_{r}\right\rangle \\&=\left\langle \lambda _{1}v_{1},v_{1}\right\rangle +\cdots +\left\langle \lambda _{r}v_{r},v_{r}\right\rangle \\&=\lambda _{1}\left\langle v_{1},v_{1}\right\rangle +\cdots +\lambda _{r}\left\langle v_{r},v_{r}\right\rangle \\&>0.\end{aligned}}$

This means that on this linear subspace, the restricted form is positive definite; hence,

${}p\geq \sum _{i=1}^{r}\dim _{K}{\left(\operatorname {Eig} _{\lambda _{i}}{\left(\varphi \right)}\right)}\,.$

If ${}p$ were strictly larger that this dimension, there existed a ${}p$-dimensional linear subspace ${}W\subseteq V$ such that the restriction of ${}\Psi$ to ${}W$ is positive definite. Because of fact, we have

${}W\cap {\left(\operatorname {Eig} _{\lambda _{r+1}}{\left(\varphi \right)}\oplus \cdots \oplus \operatorname {Eig} _{\lambda _{s}}{\left(\varphi \right)}\oplus \operatorname {Eig} _{0}{\left(\varphi \right)}\right)}\neq 0\,.$

This yields a contradiction, since the form ${}\Psi$ is negative semidefinite on the right-hand space. Therefore,

${}p=\sum _{i=1}^{r}\dim _{K}{\left(\operatorname {Eig} _{\lambda _{i}}{\left(\varphi \right)}\right)}\,.$

The argument for ${}q$ is the same.

To fact