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)
“It is very hard to give a just definition of love. The most we can say of it is this: that in the soul, it is a desire to rule; in the spirit, it is a sympathy; and in the body, it is but a hidden and subtle desire to possessafter many mysterieswhat one loves.”
—François, Duc De La Rochefoucauld (16131680)