Block matrices and symmetric perturbations

We prove that if A=[A_ij]∈R^N,N is a block symmetric matrix and y is a solution of a nearby linear system (A+E)y=b, then there exists F=F^T such that y solves a nearby symmetric system (A+F)y=b, if A is symmetric positive definite or the matricial norm μ(A)=(‖A_ij‖₂) is diagonally dominant. Our blockwise analysis extends existing normwise and componentwise results on preserving symmetric perturbations (cf. [J.R. Bunch, J.W. Demmel, Ch. F. Van Loan, The strong stability of algorithms for solving symmetric linear systems, SIAM J.Matrix Anal. Appl. 10 (4) (1989) 494-499; D. Herceg, N. Kreji?, On the strong componentwise stability and H-matrices, Demonstratio Mathematica 30 (2) (1997) 373-378; A. Smoktunowicz, A note on the strong componentwise stability of algorithms for solving symmetric linear systems, Demonstratio Mathematica 28 (2) (1995) 443-448]).