Quadratic and Bilinear Forms

Abstract

Quadratic and bilinear forms on vector spaces are considered. A connection between the notion of bilinear form and that of linear transformation is established, based on the isomorphism between the space of bilinear forms and the space of linear transformations of the vector space to the dual space. A theorem on reducing a quadratic form to canonical form is proved, and the corresponding normal forms for symmetric and antisymmetric bilinear forms are established. Complex, real, and Hermitian forms are investigated in greater detail. For illustration of the obtained results, we consider an application of Sylvester’s criterion to algebraic equations (necessary and sufficient conditions for a real polynomial to have only real roots).