The nonunique parallel sum

This paper is a sequel to earlier study of the authors' on the nonunique shorted matrix under the failure of regularity conditions. When A and B are real symmetric nonnegative definite (n.n.d.) matrices of the same order, Anderson and Trapp present an alternative definition of the parallel sum of Anderson and Duffin which uses a shorted version of the matrix Formula Represented Here, Λ being real symmetric n.n.d., the regularity conditions are trivially true. In the more general case, however, a similar approach based on the nonunique shorted matrix leads to a nonunique parallel sum. The extent to which the nonunique parallel sum retains the properties of parallel sum of Anderson and Duffin is examined. An interesting statistical interpretation of the parallel sum operation is provided in an appendix.