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:
“As all historians know, the past is a great darkness, and filled with echoes. Voices may reach us from it; but what they say to us is imbued with the obscurity of the matrix out of which they come; and try as we may, we cannot always decipher them precisely in the clearer light of our day.”
—Margaret Atwood (b. 1939)
“According to our social pyramid, all men who feel displaced racially, culturally, and/or because of economic hardships will turn on those whom they feel they can order and humiliate, usually women, children, and animals—just as they have been ordered and humiliated by those privileged few who are in power. However, this definition does not explain why there are privileged men who behave this way toward women.”
—Ana Castillo (b. 1953)