On parallel sum and difference of matrices

Abstract

Anderson and Duffin [l] h ave recently introduced the concept of parallel sum of a pair of matrices and established interesting and important properties of this operation when the matrices concerned are nonnegative definite (n.n.d.). Applications of this concept in Electrical Network Theory were discussed by the same authors and in some statistical problems connected with the estimation of parameters in a dynamic linear system by Anderson et al. [3]. The concept of parallel sum was extended and its elegance further demonstrated by Rao and Mitra [7], who showed that most of the properties proved by Anderson and Duffin [l] are indeed true for a much wider class of pairs of matrices designated by these authors as “parallel summable.” The object of the present paper is to explore additional properties of the parallel sum to obtain the condition for consistency together with the complete class of solutions (X) of the parallel sum equation