applied to a single vector is always a real number. One can show that a complex sesquilinear form is Hermitian if and only if the associated quadratic form is real for all Skew-Hermitian form A complex skew-Hermitian form (also called an antisymmetric sesquilinear form), is a complex sesquilinear form such that Every complex skew-Hermitian form can be written as the imaginary unit times a Hermitian form. The matrix representation of a complex skew-Hermitian form is a skew-Hermitian matrix. A complex skew-Hermitian form applied to a single vector is always a purely imaginary number. Over a division ring This section applies unchanged when the division ring is commutative. More specific terminology then also applies: the division ring is a field, the anti-automorphism is also an automorphism, and the right module is a vector space. The following applies to a left module with suitable reordering of expressions. Definition A -sesquilinear form over a right -module is a bi-additive map with an associated anti-automorphism of a division ring such that, for all in and all in , The associated anti-automorphism for any nonzero sesquilinear form is uniquely determined by . Orthogonality Given a sesquilinear form over a module and a subspace (submodule) of , the orthogonal complement of with respect to is Similarly, is orthogonal to with respect to , written (or simply if can be inferred from the context), when . This relation need not be symmetric, i.e. does not imply (but see below). Reflexivity A sesquilinear form is reflexive if, for all in , implies That is, a sesquilinear form is reflexive precisely when the derived orthogonality relation is symmetric. Hermitian variations A -sesquilinear form is called -Hermitian if there exists in such that, for all in , If , the form is called -Hermitian, and if , it is called -anti-Hermitian. (When is implied, respectively simply Hermitian or anti-Hermitian.) For a nonzero -Hermitian form, it follows that for all in , It also follows that is a fixed point of the map . The fixed points of this map form a subgroup of the additive group of . A -Hermitian form is reflexive, and every reflexive -sesquilinear form is -Hermitian for some . In the special case that is the identity map (i.e., ), is commutative, is a bilinear form and . Then for the bilinear form is called symmetric, and for is called skew-symmetric. Example Let be the three