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:

    “And into the gulf between cantankerous reality and the male ideal of shaping your world, sail the innocent children. They are right there in front of us—wild, irresponsible symbols of everything else we can’t control.”
    —Hugh O’Neill (20th century)

    “Just as a new scientific discovery manifests something that was already latent in the order of nature, and at the same time is logically related to the total structure of the existing science, so the new poem manifests something that was already latent in the order of words.”
    —Northrop Frye (b. 1912)