Inversion of Matrices by Partitioning

Abstract

The inversion of nonsingular matrices is considered. A method is developed which starts with an arbitrary partitioning of the given matrix. The separate submatrices are grouped into sets determined by the nonzero entries of some appropriate group, G , of permutation matrices. The group structure of G then establishes a sequence of operations on these sets of submatrices from which the corresponding representation of the inverse is obtained. Whether the method described is to be preferred to, say, Gauss's algorithm will depend on the capabilities that are required by other parts of the algorithm that is to be implemented in the special-purpose parallel computer. The basic speed, measured by the count of parallel multiplications and divisions, is comparable to that obtained with Gauss's algorithm and is slightly better under certain conditions. The principal difference is that this method uses primarily matrix multiplication, whereas Gauss's algorithm uses primarily row combinations. When the special-purpose computer under design must supply this capability anyway, the method developed here should be considered. Application of the process is limited to matrices for which we can set up a partitioning such that we can guarantee, a priori, that certain of the submatrices are nonsingular. Hence the method is not useful for arbitrary nonsingular matrices. However, it can be applied to certain important classes of matrices, notably those that are “dominated by the diagonal.” Noise covariance matrices are of this type; therefore the method can be applied to them. The inversion of a noise covariance matrix is required in some problems of optimal prediction and control. It is for applications of this sort that the method seems particularly attractive.