dimensional vector space over the finite field , where is a prime power. With respect to the standard basis we can write and and define the map by: The map is an involutory automorphism of . The map is then a -sesquilinear form. The matrix associated to this form is the identity matrix. This is a Hermitian form. In projective geometry Assumption: In this section, sesquilinear forms are antilinear (resp. linear) in their second (resp. first) argument. In a projective geometry , a permutation of the subspaces that inverts inclusion, i.e. for all subspaces , of , is called a correlation. A result of Birkhoff and von Neumann (1936) shows that the correlations of desarguesian projective geometries correspond to the nondegenerate sesquilinear forms on the underlying vector space. A sesquilinear form is nondegenerate if for all in (if and) only if . To achieve full generality of this statement, and since every desarguesian projective geometry may be coordinatized by a division ring, Reinhold Baer extended the definition of a sesquilinear form to a division ring, which requires replacing vector spaces by -modules. (In the geometric literature these are still referred to as either left or right vector spaces over skewfields.) Over arbitrary rings The specialization of the above section to skewfields was a consequence of the application to projective geometry, and not intrinsic to the nature of sesquilinear forms. Only the minor modifications needed to take into account the non-commutativity of multiplication are required to generalize the arbitrary field version of the definition to arbitrary rings. Let be a ring, an -module and an antiautomorphism of . A map is -sesquilinear if for all in and all in . An element is orthogonal to another element with respect to the sesquilinear form (written ) if . This relation need not be symmetric, i.e. does not imply . A sesquilinear form is reflexive (or orthosymmetric) if implies for all in . A sesquilinear form is Hermitian if there exists such that for all in . A Hermitian form is necessarily reflexive, and if it is nonzero, the associated antiautomorphism is an involution (i.e. of order 2). Since for an antiautomorphism we have for all in , if , then must be commutative and is a bilinear form. In particular, if, in this case, is a skewfield, then is a field and is a vector space with a bilinear form. An antiautomorphism