Definition of "symmetric bilinear (real) form indefinite"

Question

In my studies I use these definition:

Def.: $f \in \mathscr{B}_ \Bbb{R}((e \times e), \Bbb{R}) $, $f $ is symmetric bilinear (real) form positive definite if
1) $\forall x \in e(f(x,x)\geq0)$
2) $\forall x \in e(f(x,x)=0 \leftrightarrow x=0)$

Def.: $f \in \mathscr{B}_ \Bbb{R}((e \times e), \Bbb{R}) $, $f $ is symmetric bilinear (real) form negative definite if
1) $\forall x \in e(f(x,x)\leq0)$
2) $\forall x \in e(f(x,x)=0 \leftrightarrow x=0)$

Def.: $f \in \mathscr{B}_ \Bbb{R}((e \times e), \Bbb{R}) $, $f $ is symmetric bilinear (real) form postive semi-definite if
1) $\forall x \in e(f(x,x)\geq0)$
2) $\exists z \in e(z \neq 0 \wedge f(z,z)=0 )$

Def.: $f \in \mathscr{B}_ \Bbb{R}((e \times e), \Bbb{R}) $, $f $ is symmetric bilinear (real) form negative semi-definite if
1) $\forall x \in e(f(x,x)\leq0)$
2) $\exists z \in e(z \neq 0 \wedge f(z,z)=0 )$

I need definition of "symmetric bilinear (real) form indefinite"...

Thanks in advance!

Answer 1

It is indefinite if it is neither of the above. For example, in $\mathbb R^2 \times \mathbb R^2$: $$x^T\begin{bmatrix} 1&0\\0&-1 \end{bmatrix}x.$$