In mathematics, a reality structure on a complex vector space V is a decomposition of V into two real subspaces, called the real and imaginary parts of V:
Here VR is a real subspace of V, i.e. a subspace of V considered as a vector space over the real numbers. If V has complex dimension n (real dimension 2n), then VR must have real dimension n.
The standard reality structure on the vector space is the decomposition
In the presence of a reality structure, every vector in V has a real part and an imaginary part, each of which is a vector in VR:
In this case, the complex conjugate of a vector v is defined as follows:
This map is an antilinear involution, i.e.
Conversely, given an antilinear involution on a complex vector space V, it is possible to define a reality structure on V as follows. Let
and define
Then
This is actually the decomposition of V as the eigenspaces of the real linear operator c. The eigenvalues of c are +1 and −1, with eigenspaces VR and VR, respectively. Typically, the operator c itself, rather than the eigenspace decomposition it entails, is referred to as the reality structure on V.
Read more about Reality Structure: See Also
Famous quotes containing the words reality and/or structure:
“Most young black females learn to be suspicious and critical of feminist thinking long before they have any clear understanding of its theory and politics.... Without rigorously engaging feminist thought, they insist that racial separatism works best. This attitude is dangerous. It not only erases the reality of common female experience as a basis for academic study; it also constructs a framework in which differences cannot be examined comparatively.”
—bell hooks (b. c. 1955)
“With sixty staring me in the face, I have developed inflammation of the sentence structure and definite hardening of the paragraphs.”
—James Thurber (1894–1961)