Linear spaces and preservers of persymmetric triangular matrices of bounded rank-two / Ng Zhen Chuan - Ng, Zhen Chuan (2012)
Let F be a field and n an integer > 2. We say that a square matrix A is persymmetric if A is symmetric in the second diagonal. Let STn(F) denote the linear space of all n x n persymmetric upper triangular matrices over F. A subspace S of STn(F) is said to be a space of bounded rank-two matrices if each matrix in S has rank bounded above by two, and a rank-two space if each nonzero element in it has rank two. In this dissertation, we classify subspaces of bounded rank-two matrices of STn(F) over a field F with at least three elements. As a corollary, a complete description of rank-two subspaces of STn(F) is obtained. We next deduce from the structural results of subspaces of bounded rank-two matrices of STn(F), a characterization of linear maps : STn(F) ! STm(F), m > n > 2, that send nonzero matrices with rank at most two to nonzero matrices with rank at most two.