View source for Hermitian form

''on a left <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/h/h047/h047050/h0470502.png" />-module <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/h/h047/h047050/h0470503.png" />''

A mapping <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/h/h047/h047050/h0470504.png" /> that is linear in the first argument and satisfies the condition

<table class="eq" style="width:100%;"> <tr><td valign="top" style="width:94%;text-align:center;"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/h/h047/h047050/h0470505.png" /></td> </tr></table>

Here <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/h/h047/h047050/h0470506.png" /> is a ring with a unit element and equipped with an involutory anti-automorphism <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/h/h047/h047050/h0470507.png" />. In particular, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/h/h047/h047050/h0470508.png" /> is a [[Sesquilinear form|sesquilinear form]] on <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/h/h047/h047050/h0470509.png" />. The module <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/h/h047/h047050/h04705010.png" /> 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 <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/h/h047/h047050/h04705011.png" /> form a group <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/h/h047/h047050/h04705012.png" />, which is called the unitary group associated with the Hermitian form <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/h/h047/h047050/h04705013.png" />; its structure has been well studied when <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/h/h047/h047050/h04705014.png" /> is a skew-field (see [[Unitary group|Unitary group]]).

A Hermitian form is a special case of an <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/h/h047/h047050/h04705016.png" />-Hermitian form (where <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/h/h047/h047050/h04705017.png" /> is an element in the centre of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/h/h047/h047050/h04705018.png" />), that is, a sesquilinear form <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/h/h047/h047050/h04705019.png" /> on <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/h/h047/h047050/h04705020.png" /> for which

<table class="eq" style="width:100%;"> <tr><td valign="top" style="width:94%;text-align:center;"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/h/h047/h047050/h04705021.png" /></td> </tr></table>

When <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/h/h047/h047050/h04705022.png" />, an <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/h/h047/h047050/h04705023.png" />-Hermitian form is Hermitian, and when <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/h/h047/h047050/h04705024.png" /> the form is called skew-Hermitian or anti-Hermitian. If <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/h/h047/h047050/h04705025.png" />, 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

<table class="eq" style="width:100%;"> <tr><td valign="top" style="width:94%;text-align:center;"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/h/h047/h047050/h04705026.png" /></td> </tr></table>

where <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/h/h047/h047050/h04705027.png" /> for any <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/h/h047/h047050/h04705028.png" />, is bijective, then <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/h/h047/h047050/h04705029.png" /> is called a non-degenerate Hermitian form or a Hermitian scalar product on <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/h/h047/h047050/h04705030.png" />.

If <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/h/h047/h047050/h04705031.png" /> is a free <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/h/h047/h047050/h04705032.png" />-module with a basis <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/h/h047/h047050/h04705033.png" />, then the matrix <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/h/h047/h047050/h04705034.png" />, where <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/h/h047/h047050/h04705035.png" />, is called the matrix of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/h/h047/h047050/h04705036.png" /> in the given basis; it is a [[Hermitian matrix|Hermitian matrix]] (that is, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/h/h047/h047050/h04705037.png" />). A Hermitian form <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/h/h047/h047050/h04705038.png" /> is non-degenerate if and only if <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/h/h047/h047050/h04705039.png" /> is invertible. If <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/h/h047/h047050/h04705040.png" /> is a skew-field, if <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/h/h047/h047050/h04705041.png" />, and if <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/h/h047/h047050/h04705042.png" /> is finite-dimensional over <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/h/h047/h047050/h04705043.png" />, then <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/h/h047/h047050/h04705044.png" /> has an orthogonal basis relative to <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/h/h047/h047050/h04705045.png" /> (in which the matrix is diagonal).

If <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/h/h047/h047050/h04705046.png" /> is a commutative ring with identity, if <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/h/h047/h047050/h04705047.png" />, and if the matrix of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/h/h047/h047050/h04705048.png" /> is definite, then its determinant lies in <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/h/h047/h047050/h04705049.png" />. Under a change of basis in <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/h/h047/h047050/h04705050.png" /> this determinant is multiplied by a non-zero element of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/h/h047/h047050/h04705051.png" /> of the form <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/h/h047/h047050/h04705052.png" />, where <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/h/h047/h047050/h04705053.png" /> is an invertible element of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/h/h047/h047050/h04705054.png" />. The determinant regarded up to multiplication by such elements is called the determinant of the Hermitian form or of the Hermitian space <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/h/h047/h047050/h04705055.png" />; it is an important invariant and is used in the classification of Hermitian forms.

Let <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/h/h047/h047050/h04705056.png" /> be commutative. Then a Hermitian form <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/h/h047/h047050/h04705057.png" /> on <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/h/h047/h047050/h04705058.png" /> gives rise to a quadratic form <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/h/h047/h047050/h04705059.png" /> on <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/h/h047/h047050/h04705060.png" /> over <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/h/h047/h047050/h04705061.png" />. The analysis of such forms lies at the basis of the construction of the Witt group of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/h/h047/h047050/h04705062.png" /> with an involution (see [[Witt ring|Witt ring]]; [[Witt decomposition|Witt decomposition]]; [[Witt theorem|Witt theorem]]). When <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/h/h047/h047050/h04705063.png" /> is a maximal ordered field, then the [[Law of inertia|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 <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/h/h047/h047050/h04705064.png" /> is a field and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/h/h047/h047050/h04705065.png" />, then <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/h/h047/h047050/h04705066.png" /> is a quadratic Galois extension of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/h/h047/h047050/h04705067.png" />, and isometry of two non-degenerate Hermitian forms over <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/h/h047/h047050/h04705068.png" /> is equivalent to isometry of the quadratic forms over <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/h/h047/h047050/h04705069.png" /> generated by them; this reduces the classification of non-degenerate Hermitian forms over <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/h/h047/h047050/h04705070.png" /> to that of non-degenerate quadratic forms over <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/h/h047/h047050/h04705071.png" />.

If <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/h/h047/h047050/h04705072.png" /> and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/h/h047/h047050/h04705073.png" /> 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 <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/h/h047/h047050/h04705074.png" /> 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 <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/h/h047/h047050/h04705075.png" /> is a finite field, then there is only one invariant, the rank. For the case when <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/h/h047/h047050/h04705076.png" /> is an algebraic extension of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/h/h047/h047050/h04705077.png" />, see [[#References|[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====
<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)</TD></TR><TR><TD valign="top">[2]</TD> <TD valign="top">  J.A. Dieudonné,   "La géométrie des groups classiques" , Springer  (1955)</TD></TR><TR><TD valign="top">[3]</TD> <TD valign="top">  J. Milnor,   D. Husemoller,   "Symmetric bilinear forms" , Springer  (1973)</TD></TR><TR><TD valign="top">[4]</TD> <TD valign="top">  O.T. O'Meara,   "Introduction to quadratic forms" , Springer  (1973)</TD></TR></table>