Hermitian form

on a left -module

A mapping that is linear in the first argument and satisfies the condition

Here is a ring with a unit element and equipped with an involutory anti-automorphism . In particular, is a sesquilinear form on . The module itself is then called a Hermitian space. By analogy with what is done for bilinear forms, equivalence is defined for Hermitian forms (in another terminology, isometry) and, correspondingly, isomorphism (isometry) of Hermitian spaces (in particular, automorphism). All automorphisms of a Hermitian form form a group , which is called the unitary group associated with the Hermitian form ; its structure has been well studied when is a skew-field (see Unitary group).

A Hermitian form is a special case of an -Hermitian form (where is an element in the centre of ), that is, a sesquilinear form on for which

When , an -Hermitian form is Hermitian, and when the form is called skew-Hermitian or anti-Hermitian. If , a Hermitian form is a symmetric bilinear form, and a skew-Hermitian form is a skew-symmetric or anti-symmetric bilinear form. If the mapping

where for any , is bijective, then is called a non-degenerate Hermitian form or a Hermitian scalar product on .

If is a free -module with a basis , then the matrix , where , is called the matrix of in the given basis; it is a Hermitian matrix (that is, ). A Hermitian form is non-degenerate if and only if is invertible. If is a skew-field, if , and if is finite-dimensional over , then has an orthogonal basis relative to (in which the matrix is diagonal).

If is a commutative ring with identity, if , and if the matrix of is definite, then its determinant lies in . Under a change of basis in this determinant is multiplied by a non-zero element of of the form , where is an invertible element of . The determinant regarded up to multiplication by such elements is called the determinant of the Hermitian form or of the Hermitian space ; it is an important invariant and is used in the classification of Hermitian forms.

Let be commutative. Then a Hermitian form on gives rise to a quadratic form on over . The analysis of such forms lies at the basis of the construction of the Witt group of with an involution (see Witt ring; Witt decomposition; Witt theorem). When is a maximal ordered field, then the law of inertia extends to Hermitian forms (and there arise the corresponding concepts of the signature, the index of inertia, and positive and negative definiteness). If is a field and , then is a quadratic Galois extension of , and isometry of two non-degenerate Hermitian forms over is equivalent to isometry of the quadratic forms over generated by them; this reduces the classification of non-degenerate Hermitian forms over to that of non-degenerate quadratic forms over .

If and is the involution of complex conjugation, then a complete system of invariants of Hermitian forms over a finite-dimensional space is given by the rank and the signature of the corresponding quadratic forms. If is a local field or the field of functions of a single variable over a finite field, then a complete system of invariants for non-degenerate Hermitian forms is given by the rank and the determinant. If is a finite field, then there is only one invariant, the rank. For the case when is an algebraic extension of , see [3]. Ch. Hermite was the first, in 1853, to consider the forms that bear his name in connection with certain problems of number theory.

References

[1]	N. Bourbaki, "Elements of mathematics. Algebra: Algebraic structures. Linear algebra" , 1 , Addison-Wesley (1974) pp. Chapt.1;2 (Translated from French) MR0354207
[2]	J.A. Dieudonné, "La géométrie des groups classiques" , Springer (1955) Zbl 0221.20056
[3]	J. Milnor, D. Husemoller, "Symmetric bilinear forms" , Springer (1973) MR0506372 Zbl 0292.10016
[4]	O.T. O'Meara, "Introduction to quadratic forms" , Springer (1973) Zbl 0259.10018