Not degenerated bilinear form

Definitions

That is to say $E$ a vector Space on a body $K$. That is to say $f$ a bilinear Form on $E$.

• One calls singular space of $f$ (one says also core) the vectorial Sous-espace of $E$ according to:

$S\_g\; (F)\; =\; \backslash \; \{X\; \backslash \; in\; E,\; \backslash \; \backslash \; forall\; there\; \backslash \; in\; E,\; \backslash \; F\; (X,\; there)\; =0\; \backslash \}$

• When $f$ is symmetrical, this definition is sufficient. If not, one is brought to define as singular space on the right (core on the right) following space:

$S\_d\; (F)\; =\; \backslash \; \{X\; \backslash \; in\; E,\; \backslash \; \backslash \; forall\; there\; \backslash \; in\; E,\; \backslash \; F\; (there,\; X)\; =0\; \backslash \}$

• One says that $f$ is not degenerated if and only if $S\_g\; (F)\; =\; S\_d\; (F)\; =\; \backslash \; \{\backslash \; overrightarrow\; \{0\}\; \backslash \}$.

Properties

• For a vector $x$ of $E$, let us note $f\; (X.)$ the Function partial of $f$ which with $y$ associates $f\; (X,\; there)$. It is a linear form. Moreover, the application $\backslash \; hat\; \{F\}$ of $E$ in $E^*$ (dual Espace of E) which with $x$ associates $f\; (X.)$ is linear.

By construction $S\_g\; (F)\; =\; Ker\; \backslash \; hat\; \{F\}$.

• In finished dimension $S\_g\; (F)\; =\; \backslash \; \{\backslash \; overrightarrow\; \{0\}\; \backslash \}$ if and only if $S\_d\; (F)\; =\; \backslash \; \{\backslash \; overrightarrow\; \{0\}\; \backslash \}$.

• When $E$ is a real vector space, any symmetrical bilinear form not degenerated positive on $E$ is strictly positive (it is a scalar Produit).

It is a consequence of the Inégalité of Cauchy-Schwarz for the positive bilinear forms.

References

• J.M. Arnaudiès and H. Fraysse Course of mathematics 4: Bilinear algebra and geometry 1990 Dunod

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