the other convention and take the first argument to be conjugate-linear (i.e. antilinear) and the second to be linear. This is the convention used mostly by physicists and originates in Dirac's bra–ket notation in quantum mechanics. In the more general noncommutative setting, with right modules we take the second argument to be linear and with left modules we take the first argument to be linear. Complex vector spaces Assumption: In this section, sesquilinear forms are antilinear in their first argument and linear in their second. Over a complex vector space a map is sesquilinear if for all and all Here, is the complex conjugate of a scalar A complex sesquilinear form can also be viewed as a complex bilinear map where is the complex conjugate vector space to By the universal property of tensor products these are in one-to-one correspondence with complex linear maps For a fixed the map is a linear functional on (i.e. an element of the dual space ). Likewise, the map is a conjugate-linear functional on Given any complex sesquilinear form on we can define a second complex sesquilinear form via the conjugate transpose: In general, and will be different. If they are the same then is said to be . If they are negatives of one another, then is said to be . Every sesquilinear form can be written as a sum of a Hermitian form and a skew-Hermitian form. Matrix representation If is a finite-dimensional complex vector space, then relative to any basis of a sesquilinear form is represented by a matrix and given by where is the conjugate transpose. The components of the matrix are given by Hermitian form The term Hermitian form may also refer to a different concept than that explained below: it may refer to a certain differential form on a Hermitian manifold. A complex Hermitian form (also called a symmetric sesquilinear form), is a sesquilinear form such that The standard Hermitian form on is given (again, using the "physics" convention of linearity in the second and conjugate linearity in the first variable) by More generally, the inner product on any complex Hilbert space is a Hermitian form. A minus sign is introduced in the Hermitian form to define the group SU(1,1). A vector space with a Hermitian form is called a Hermitian space. The matrix representation of a complex Hermitian form is a Hermitian matrix. A complex Hermitian form