A bijective mapping between projective spaces of the same finite dimension such that implies . The image of a sum of subspaces under a correlation is the intersection of their images and, conversely, the image of an intersection is the sum of the images. In particular, the image of a point is a hyperplane and vice versa. A necessary and sufficient condition for the existence of a correlation of a projective space over a division ring onto a space over a division ring is that there exists an anti-isomorphism , i.e. a bijective mapping for which , ; in that case is dual to . Examples of spaces with an auto-correlation, i.e. a correlation onto itself, are the real projective spaces , the complex projective spaces and the quaternion projective spaces .
A polarity is an auto-correlation satisfying . A projective space over a division ring admits a polarity if and only if admits an involutory anti-automorphism, i.e. an anti-automorphism with .
A subspace is called a null subspace relative to an auto-correlation if for any point , and strictly isotropic if . Any strictly isotropic subspace is a null subspace. A polarity relative to which the whole space is a null space is called a null (or symplectic) polarity (see also Polarity).
Let the projective space over a division ring be interpreted as the set of linear subspaces of the (left) linear space over . A semi-bilinear form on is a mapping together with an anti-automorphism of such that
In particular, if is a field and , then is a bilinear form. A semi-bilinear form is called non-degenerate provided for all (all ) implies (, respectively). Any auto-correlation of can be represented with the aid of a non-degenerate semi-bilinear form in the following way: for a subspace of its image is the orthogonal complement of with respect to :
(the Birkhoff–von Neumann theorem, ). is a polarity if and only if is reflexive, i.e. if implies . By multiplying by a suitable element of one can bring any reflexive non-degenerate semi-bilinear form and the corresponding automorphism in either of the following two forms:
1) is an involution, i.e. , and
In this case one calls symmetric if (and hence necessarily is a field) and Hermitian if .
2) (and hence is a field) and
Such an is called anti-symmetric.
A special example of a correlation is the following. Let be a projective space over a division ring . Define the opposite division ring as the set of elements of with the same addition but with multiplication
is an anti-isomorphism from onto which defines the canonical correlation from onto . The (left) projective space , which can be identified with the right projective space , i.e. with the set of linear subspaces of the -dimensional right vector space , is the (canonical) dual space of (cf. Projective algebra, the construction of ).
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Correlation. Encyclopedia of Mathematics. URL: http://encyclopediaofmath.org/index.php?title=Correlation&oldid=13946