Sesquilinear form | Wikipedia audio article
Channel: wikipedia tts
Date: 2019-05-06

This is an audio version of the Wikipedia Article:
https://en.wikipedia.org/wiki/Sesquilinear_form


00:01:23 1 Convention
00:02:10 2 Complex vector spaces
00:04:16 2.1 Matrix representation
00:05:54 2.2 Hermitian form
00:05:57 2.3 Skew-Hermitian form
00:07:58 3 Over a division ring
00:09:04 3.1 Definition
00:09:33 3.2 Orthogonality
00:10:27 3.3 Reflexivity
00:11:26 3.4 Hermitian variations
00:11:36 3.5 Example
00:12:23 4 In projective geometry
00:13:01 5 Over arbitrary rings
00:14:26 6 See also
00:14:40 7 Notes
00:14:56 8 References
00:15:08 9 External links
00:16:28 In projective geometry
00:17:06 0 for all y in V (if and) only if x
00:17:42 Over arbitrary rings
00:19:54 0 implies φ(y, x)
00:20:44 σ(t) σ(s) for all s, t in R, if σ
00:21:21 ba, where the product on the right is the product in R. It follows from this that a right (left) R-module V can be turned into a left (right) Rop-module, Vo. Thus, the sesquilinear form φ : V × V → R can be viewed as a bilinear form φ′ : V × Vo → R.
00:21:39 See also
00:21:48 *-ring



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SUMMARY
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In mathematics, a sesquilinear form is a generalization of a bilinear form that, in turn, is a generalization of the concept of the dot product of Euclidean space. A bilinear form is linear in each of its arguments, but a sesquilinear form allows one of the arguments to be "twisted" in a semilinear manner, thus the name; which originates from the Latin numerical prefix sesqui- meaning "one and a half". The basic concept of the dot product – producing a scalar from a pair of vectors – can be generalized by allowing a broader range of scalar values and, perhaps simultaneously, by widening the definition of what a vector is.
A motivating special case is a sesquilinear form on a complex vector space, V. This is a map V × V → C that is linear in one argument and "twists" the linearity of the other argument by complex conjugation (referred to as being antilinear in the other argument). This case arises naturally in mathematical physics applications. Another important case allows the scalars to come from any field and the twist is provided by a field automorphism.
An application in projective geometry requires that the scalars come from a division ring (skewfield), K, and this means that the "vectors" should be replaced by elements of a K-module. In a very general setting, sesquilinear forms can be defined over R-modules for arbitrary rings R.