On the Inverse of a Paferned Covariance Matrix

In the following discussion, AT is the transpose of a matrix A, and D[Ai] is a diagonal (or block-diagonal) matrix whose ith element (or diagonal matrix) is Ai. Suppose that V is an N x N covariance matrix to be inverted. V can be decomposed as D[ri]R D[rj], where R is the N x N correlation matrix and D[ri] is the N x N diagonal matrix of standard deviations. Then V -l = D[1/rj]R-D[1/ou]. Now suppose that R can be partitioned into k2 submatrices representing k classes such that within class i intraclass correlations (cii) are equal and between classes i andj the interclass correlations (cij) are equal (i, j = 1, 2, . . . , k). Let ni be the number of variates in class i, where ni + n2 + + nk = N, and take 1i to be an identity matrix of order ni, 1i to be a column vector of ni l's, and Jij to be an ni x nj matrix of l's. The matrix R can be written as