Relative and Absolute Perturbation Bounds for Weighted Polar Decomposition

Let Cm×n, Cm×n r , C m ≥ , C m > , and In denote the set of m × n complex matrices, subset of Cm×n consisting of matrices with rank r, set of the Hermitian nonnegative definite matrices of order m, subset of C ≥ consisting of positive-definite matrices and n × n unit matrix, respectively. Without specification, we always assume that m > n >max{r, s} and the given weight matrices M ∈ C > ,N ∈ C >. For A ∈ Cm×n, we denote by R A , r A , A∗, AMN N−1A∗M,AMN, ‖A‖ and ‖A‖F the column space, rank, conjugate transpose, weighted conjugate transpose or adjoint , weighted Moore-Penrose inverse, unitarily invariant norm, and Frobenius norm of A, respectively. The definitions of AMN and A † MN can be found in details in 1, 2 . The weighted polar decomposition MN-WPD of A ∈ Cm×n is given by