linear complex structure

A on a real vector space V, with dim(V)=m, is a linear automorphismPlanetmathPlanetmathPlanetmath JAut(V) such that J2=JJ=-idV. With a complex structure J we can consider V as a complex vector space with the productMathworldPlanetmath ×VV given by


This implies that the dimensionMathworldPlanetmathPlanetmath m of V must be even.

A common example is V=2n with the standard basis 𝐞1,,𝐞n,𝐟1,,𝐟n, for which we can obtain a complex structure J0Aut(2n) represented by the matrix


Here 𝐈nMn() is the identityPlanetmathPlanetmath n×n matrix and 𝟎Mn() is the zero n×n matrix.

Title linear complex structure
Canonical name LinearComplexStructure
Date of creation 2013-03-22 16:16:48
Last modified on 2013-03-22 16:16:48
Owner Mazzu (14365)
Last modified by Mazzu (14365)
Numerical id 12
Author Mazzu (14365)
Entry type Definition
Classification msc 15-00
Related topic ComplexificationOfVectorSpace
Defines linear complex structure