Difference between revisions of "Hermitian form"
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− | <table><TR><TD valign="top">[1]</TD> <TD valign="top"> | + | <table><TR><TD valign="top">[1]</TD> <TD valign="top"> N. Bourbaki, "Elements of mathematics. Algebra: Algebraic structures. Linear algebra" , '''1''' , Addison-Wesley (1974) pp. Chapt.1;2 (Translated from French) {{MR|0354207}} {{ZBL|}} </TD></TR><TR><TD valign="top">[2]</TD> <TD valign="top"> J.A. Dieudonné, "La géométrie des groups classiques" , Springer (1955) {{MR|}} {{ZBL|0221.20056}} </TD></TR><TR><TD valign="top">[3]</TD> <TD valign="top"> J. Milnor, D. Husemoller, "Symmetric bilinear forms" , Springer (1973) {{MR|0506372}} {{ZBL|0292.10016}} </TD></TR><TR><TD valign="top">[4]</TD> <TD valign="top"> O.T. O'Meara, "Introduction to quadratic forms" , Springer (1973) {{MR|}} {{ZBL|0259.10018}} </TD></TR></table> |
Revision as of 17:33, 31 March 2012
on a left -module
A mapping that is linear in the first argument and satisfies the condition
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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
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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
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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 |
Hermitian form. Encyclopedia of Mathematics. URL: http://encyclopediaofmath.org/index.php?title=Hermitian_form&oldid=14638