Degenerate form

In mathematics, specifically linear algebra, a degenerate bilinear form f(x, y) on a vector space V is a bilinear form such that the map from V to V^∗ (the dual space of V) given by v ↦ (x ↦ f(x, v)) is not an isomorphism. An equivalent definition when V is finite-dimensional is that it has a non-trivial kernel: there exist some non-zero x in V such that

A nondegenerate or nonsingular form is one that is not degenerate, meaning that ${\displaystyle v\mapsto (x\mapsto f(x,v))}$ is an isomorphism, or equivalently in finite dimensions, if and only if

If V is finite-dimensional then, relative to some basis for V, a bilinear form is degenerate if and only if the determinant of the associated matrix is zero – if and only if the matrix is singular, and accordingly degenerate forms are also called singular forms. Likewise, a nondegenerate form is one for which the associated matrix is non-singular, and accordingly nondegenerate forms are also referred to as non-singular forms. These statements are independent of the chosen basis.

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