Hermitian Form

Hermitian form

[er′mish·ən ′fȯrm]
(mathematics)
A polynomial in n real or complex variables where the matrix constructed from its coefficients is Hermitian.
More generally, a sesquilinear form g such that g (x,y) = g (y,x)for all values of the independent variables x and y, where g (x,y)is the image of g (x,y) under the automorphism of the underlying ring.

Hermitian Form

 

an expression of the type

where akt = ātk (ā is the complex conjugate of a). A matrix constructed from the coefficients of a Hermitian form is said to be Hermitian, as is a linear transformation that is defined by a Hermitian matrix. In 1854, C. Hermite investigated the representation of whole numbers by Hermitian forms for integral values of the arguments. The theory of Hermitian forms is in many respects similar to the theory of quadratic forms.

References in periodicals archive ?
The quaternion algebra is one of the most important and well-studied objects in mathematics and physics; and it has natural Hermitian form which induces Euclidean metric [1].
A functional ([dot], [dot]) : X x X [right arrow] K is said to be a Hermitian form on X if
We use the following notations related to a given Hermitian form ([dot], [dot]) on X:
The following result concerning the functional properties of [sigma] as a function depending on the nonnegative hermitian form ([dot], [dot]) has been obtained in [?
2,1] is a nonnegative hermitian form and thus, by the above property one has,
If ([dot], [dot]) is a nonnegative Hermitian form on X, x, y [member of] X and ||y|| [not equal to] 0, then
Since, by Bessel's inequality the hermitian form ([dot], [dot])[.
then, using Hadamard's inequality, we conclude that q ([dot], [dot]) is also a nonnegative hermitian form.
Classical and recent inequalities for Hermitian forms on real or complex linear spaces are surveyed.
The following fundamental facts concerning Hermitian forms hold [?
His account relies on the basic results in the linear representations of finite groups in quadratic, symplectic, and Hermitian forms and in involutions over simple algebras, which are outlined, mainly without proofs, in the first chapter.