SERIES AND PARALLEL ADDITION OF MATRICES

Let A and B be Hermitian semi-definite matrices and let A+ denote the Moore-Penrose generalized inverse. Then we define the parallel sum of A and B by the formula A(A + B)+B and denote it by A : B. If A and B are nonsingular this reduces to A : B = (A-1 + B-1)-1 which is the well known electrical formula for addition of resistors in parallel. Then it is shown that the Hermitian semi-definite matrices form a commutative partially ordered semigroup under the parallel sum operation. Here the ordering A ≥ B means A - B is semidefinite and the following inequality holds: (A + B) : (C + D) ≥ A : C + B : D. If R(A) denotes the range of A then it is found that R(A : B) = R(A) ∩ R(B). Moreover if A and B are orthogonal projection operators then 2 A : B is the orthogonal projection on R(A) ∩ R(B). The norms are found to satisfy the inequality ∥ A : B ∥ ≤ ∥ A ∥ : ∥ B ∥. Generalization to non-Hermitian operators are also developed. © 1969.