1$.

${\EuScript{J}}$-separated linear maps {#sec:j-separat-linear}
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### ${\EuScript{J}}$-symmetrical matrixes and ${\EuScript{J}}$-selfadjoint operators {#sec:j-symmetr-matrix}

The symmetrical bilinear form defined by $(v,w)=\langle J_x
v,w\rangle$, $v,w\in E_x$ for $x\in M$ endows $E_x$ with a pseudo-Euclidean structure. Since ${\EuScript{J}}_x$ is non-degenerate, then the form $(\cdot,\cdot)$ is likewise non-degenerate and many properties of inner products are shared with symmetrical non-degenerate bilinear forms. We state some of them below.

We recall that $E^\perp:=\{v\in V: (v,w)=0
    \quad\text{for all}\quad w\in E\}$, the pseudo-orthogonal space of $E$, is defined using the bilinear form.

\[pr:propbilinear\] Let $(\cdot,\cdot):V\times V \to{{\mathbb R}}$ be a real symmetric non-degenerate bilinear form on the real finite dimensional vector space $V$.

1.  $E$ is a subspace of $V$ for which $(\cdot,\cdot)$ is non-degenerate if, and only if, $V=E\oplus E^\perp$.

2.  Every base $\{v_1,\dots,v_n\}$ of $V$ can be orthogonalized by the usual Gram-Schmidt process of Euclidean spaces, that is, there are linear combinations of the basis vectors $\{w_1,\dots, w_n\}$ such that they form a basis of $V$ and $(w_i,w_j)=0$ for $i\neq j$. Then this last base can be pseudo-normalized: letting $u_i=|(w_i,w_i)|^{-1/2}w_i$ we get $(u_i,u_j)=\pm\delta_{ij}, i,j=1,\dots,n$.

3.  There exists a maximal dimension $p$ for a subspace $P_+$ of ${\EuScript{J}}$-positive vectors and a maximal dimension $q$ for a subspace 