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
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