Matrix Definition
One can define O(p,q) as a group of matrices, just as for the classical orthogonal group O(n). The standard inner product on Rp,q is given in coordinates by the diagonal matrix:
As a quadratic form,
The group O(p,q) is then the group of a n×n matrices M (where n = p+q) such that ; as a bilinear form,
Here MT denotes the transpose of the matrix M. One can easily verify that the set of all such matrices forms a group. The inverse of M is given by
One obtains an isomorphic group (indeed, a conjugate subgroup of GL(V)) by replacing η with any symmetric matrix with p positive eigenvalues and q negative ones (such a matrix is necessarily nonsingular); equivalently, any quadratic form with signature (p,q). Diagonalizing this matrix gives a conjugation of this group with the standard group O(p,q).
Read more about this topic: Indefinite Orthogonal Group
Famous quotes containing the words matrix and/or definition:
“In all cultures, the family imprints its members with selfhood. Human experience of identity has two elements; a sense of belonging and a sense of being separate. The laboratory in which these ingredients are mixed and dispensed is the family, the matrix of identity.”
—Salvador Minuchin (20th century)
“... we all know the wag’s definition of a philanthropist: a man whose charity increases directly as the square of the distance.”
—George Eliot [Mary Ann (or Marian)