Linear preservers of rank-sum-maximum, rank, rank-subtractivity, and rank-sum-minimum on symmetric matrices

摘要

Let S-n(F) be the set of all n x n symmetric matrices over a field F. For a matrix A is an element of S-n(F), rho(A) denotes the rank of A. A pair of n x n matrices (A, B) is said to be rank-sum-maximal if rho(A + B) = rho(A) + rho(B), rank-sum-minimal if rho(A + B) = rho(A) - rho(B), and rail k-subtractive if rho(A - B) = rho(A) - rho(B). We say that a linear operator phi from S-n(F) to itself is a linear preserver of rank-sum-maximum (respectively, rank-sum-minimum and rank-subtractivity) oil S-n(F) if it preserves the set of all rank-sum-maximal (respectively, rank-sum-minimal and rank-subtractive) pairs, and or rank on S-n(F) if rho(phi(X)) = rho(X) for every X is an element of S-n(F). We first characterize the linear preservers of rank-sum-maximum on S-n(F) when F is arbitrary, and thereby, linear preservers of rank (respectively, rank-subtractivity and rank-sum-minimum) oil S-n(F) are characterized.