Perturbation estimation for the parallel sum of Hermitian positive semi-definite matrices
Wei Luo, Chuanning Song, Qingxiang Xu · Linear and Multilinear Algebra · 2018
Let Cn×n be the set of all n×n complex matrices. For any Hermitian positive semi-definite matrices A and B in Cn×n, their new common upper bound less than A+B-A:B is constructed, where (A+B)† denotes the Moore–Penrose inverse of A+B, and A:B=A(A+B)†B is the parallel sum of A and B. A factorization formula for (A+X):(B+Y)-A:B-X:Y is derived, where X,Y∈Cn×n are any Hermitian positive semi-definite perturbations of A and B, respectively. Based on the derived factorization formula and the constructed common upper bound of X and Y, some new and sharp norm upper bounds of (A+X):(B+Y)-A:B are provided. Numerical examples are also provided to illustrate the sharpness of the obtained norm upper bounds.