Reality Structure

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.

\overline{\overline v} = v,\quad \overline{v + w} = \overline{v} + \overline{w},\quad\text{and}\quad \overline{\alpha v} = \overline\alpha \, \overline{v}.

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:

    “Directors like Satyajit Ray, Rossellini, Bresson, Buñuel, Forman, Scorsese, and Spike Lee have used non-professional actors precisely in order that the people we see on the screen may be scarcely more explained than reality itself. Professionals, except fo the greatest, usually play not just the necessary role, but an explanation of the role.”
    —John Berger (b. 1926)

    “A special feature of the structure of our book is the monstrous but perfectly organic part that eavesdropping plays in it.”
    —Vladimir Nabokov (1899–1977)