Matrix Inversion, Its Interest and Application in Analysis of Data

Abstract

Matrix inversion is used in the least squares analysis of data to estimate parameters and their varianoea 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-eingular 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 ;ai ~j, with v;; =v;; 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-niJll 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 a;, ~o for I i -il ~rand a;; ~a# for all i ~j. When r ~2, the inverse is easily written in terms of BH and Vif· 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 respon&e-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.

Asymptotic linear prediction of extreme order statistics

“…However in all the three cases we have for i<=j, v~j=a~bj. The following lemma, which has been known for quite some time (see, e.g., Greenberg and Sarhan [5], p. 757), exhibits V -t for such a matrix V.…”

Criteria for positive definiteness of some band matrices

“…Studies of some special cases of (t) or related matrices have been made by several authors (see e.g. Greenberg and Sarhan [2], Ponstein [4], Rutherford [5], Todd [6] and the references there). Some simple determinative criteria for positive definiteness of (t) for all n m a y be used for the purpose of determining the propriety of applying certain iterative inversion techniques (see Martin and Wilkinson [3] and related works by these authors.…”

Chapter IV: Research Tools: Statistical Methods

“…The table prepared by Steck (1958) for computing trivariate probabilities has long been needed. Greenberg and Sarhan (1959) discussed applications of matrix inversion in the analysis of correlational data. To obtain higherorder regression coefficients, Cowden (1958) described analogues to a method by which higher-order partial correlation coefficients are calculated from those of a lower order.…”