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.
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Famous quotes containing the words reality and/or structure:
“Art is not merely an imitation of the reality of nature, but in truth a metaphysical supplement to the reality of nature, placed alongside thereof for its conquest.”
—Friedrich Nietzsche (1844–1900)
“Agnosticism is a perfectly respectable and tenable philosophical position; it is not dogmatic and makes no pronouncements about the ultimate truths of the universe. It remains open to evidence and persuasion; lacking faith, it nevertheless does not deride faith. Atheism, on the other hand, is as unyielding and dogmatic about religious belief as true believers are about heathens. It tries to use reason to demolish a structure that is not built upon reason.”
—Sydney J. Harris (1917–1986)