Matrix Inversion, Its Interest and Application in Analysis of Data

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Abstract
Matrix inversion is used in the least squares analysis of data to estimate parameters and their variances and covariances. When the data come from the analysis of variance, analysis of covariance, order statistics, or the fitting of response-surfaces, the matrix to be inverted usually falls into a structured pattern that simplifies its inversion. One class of patterned matrices is characterized by non-singular symmetrical arrangemepts in which linear combinations of the first (r – 1) rows provide the right-hand portion, starting with the elements on the principal diagonal, of the rth and remaining rows. That is: for r i j, with vij = vij for all i ≠ j. The inverses of matrices of this clase contain a non-null principal diagonal, and immediately adjacent to the principal diagonal, (r − 1) non-null superdiagonals and (r – 1) non-null subdiagonals. All other elements are zero. These inverses are called diagonal matrices of type r. That is, a matrix is diagonal of type r if aij = 0 for |i – j| r and aij = aji for all i ≠ j. When r = 2, the inverse is easily written in terms of θ1i and υif. A general procedure for obtaining the inverse when r = 3 is given. The results for r = 2 are illustrated by a problem in order statistics using the two-parameter exponential distribution. Patterned matrices also are amenable to partitioning and this is another convenient device to find the exact inverse quickly. In fitting a response-surface to data, for example, a complicated-looking matrix can be abbreviated and simplified by selective partitioning. When this has been done, the exact inverse can be found by equating the product of the matrix and its inverse to the elements of the identity matrix. This procedure is illustrated with some data from a problem in the fitting of a response-surface. When the matrix has no special pattern, as in the usual regression problem, the recommended procedure for matrix inversion is the modified square root method.