How to express a parallel sum of \(k\) matrices
The paper deals with the parallel sum of \(k\) matrices. For two Hermitian nonnegative definite matrices \(A\) and \(B\) of the same size, the parallel sum is defined by means of the Moore-Penrose inverse as: \(A:B = A(A+B)^{\dagger}B.\) A natural extension to \(k\) matrices has not a simple form to be expressed. Then an alternative expression for \(A_1:A_2:\cdots:A_k \) is given in terms of block matrices. The proof of the main result is based on some rank equalities and Moore-Penrose inverse approach. Given two positive definite matrices \(A\) and \(B\), the equality \(A:B = (A^{-1}+B^{-1})^{-1}\) holds. In the paper, a neccesary and sufficient geometrical condition is derived for the equality \(A_1:A_2:\cdots:A_k = (A_1^{\dagger} + A_2^{\dagger} + \cdots + A_k^{\dagger})^{\dagger}\) for Hermitian nonnegative definite matrices of the same order \(n\).
scientific article; zbMATH DE number 1744279
scientific article