INVERTIBILITY OF LINEAR f-ORDER PRESERVERS

摘要

Let M_(mn)(F) be the set of m x n matrices over a field F, M_n(F) = M_(nn)(F), GL_n(F) be the general linear group of order n, Λ_(mn)(F) be the set of matrices of maximal rank,Ω_(mn) (F) be the set of matrices whose rank is less than min{m, n}, and O_n(F) and U_n(F) be the groups of orthogonal and, respectively, unitary matrices of order n. The symbols A~t, A~*, rk A, Im(A), and M(A) will stand for the transpose, conjugate transpose, rank, image, and linear span of columns, respectively, of a matrix A ∈ M_(mn)(F). The symbols σ_1(A) and σ(A) will denote the maximal singular value and the set of nonzero singular values for real and complex matrices. The fields of real and complex numbers will be denoted by R and C, respectively.