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Encyclopedia > Nondegenerate

In mathematics, a degenerate bilinear form f(x,y) on a vector space V is one such that for some non-zero x in V

f(x,y) = 0 for all yV.

A nondegenerate form is one that is not degenerate. That is, f is nondegenerate iff

f(x,y) = 0 for all yV

implies that x = 0.


If f vanishes identically on all vectors it is said to be totally degenerate. Given any bilinear form f on V the set of vectors

forms a totally degenerate subspace of V. f is nondegenerate iff this subspace is trivial.


If V is finite-dimensional then, relative to some basis for V, a bilinear form is degenerate iff the determinant of the associated matrix is zero. Likewise, a nondegenerate form is one for which the associated matrix is non-singular. These statements are independent of the chosen basis.


Sometimes the words anisotropic, isotropic, totally isotropic and are used for nondegenerate, degenerate, totally degenerate respectively. Although definitions of these latter words can vary slightly between authors.


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