Wai Leong Chooi

Linear preservers on triangular matrices

Abstract Let T n ( F ) denote the vector space of all n × n upper triangular matrices over a field F . We characterize those linear transformations T on T n ( F ) that satisfy one of the following properties: (1) T preserves rank one matrices; (2) T preserves both singular and nonsingular matrices; (3) T is nonsingular and T (adj A ) = adj T ( A ) for all A ∈ T n ( F ).

Coherence invariant mappings on block triangular matrix spaces

Abstract In this paper we classify bijective mappings ψ on the space of block upper triangular matrices such that both ψ and ψ −1 preserve pairs of matrices whose differences have rank one. Applications to rank one preservers and semi-isomorphisms are considered.

Adjacency preserving maps on upper triangular matrix algebras

Abstract Let n be an integer, n ⩾3, and T n the algebra of all real n × n upper triangular matrices. We describe the general form of maps on T n that preserve the adjacency in both directions. We apply this result to characterize the semigroup of injective continuous maps on T n preserving the adjacency in one direction only.

Some linear preserver problems on block triangular matrix algebras

In this article we classify rank-one nonincreasing maps ψ on n-square block triangular matrix algebras with the assumption that ψ(In) is of rank n. As applications, we obtain complete classifications of adjugate-commuting maps, and compound-commuting maps on block triangular matrix algebras.

Linear spaces and preservers of symmetric matrices of bounded rank-two

Let 𝔽 be a field of characteristic two. Let S n (𝔽) denote the vector space of all n × n symmetric matrices over 𝔽. We characterize i. subspaces of S n (𝔽) all whose elements have rank at most two where n ⩾ 3, ii. linear maps from S m (𝔽) to S n (𝔽) that sends matrices of rank at most two into matrices of rank at most two where m, n ⩾ 3 and |𝔽| ≠ 2.

Rank-one non-increasing additive maps between block triangular matrix algebras

Let m and n be positive integers with n ≤ m. In this note, we characterize rank-one non-increasing additive maps ψ from n-square block triangular matrix algebras to m-square matrix algebras over an arbitrary field with the assumption that ψ(In ) is of rank n. We deduce from this result a complete classification of adjugate-commuting additive maps between block triangular matrix algebras. As a corollary, we characterize adjugate-commuting additive maps between square matrix algebras.

Additive maps of rank r tensors and symmetric tensors

Abstract Let and be fields and let n be a positive integer. Let and be linear spaces over such that and let and be linear spaces over . Let be the tensor product of and and let be the subspace of symmetric tensors of . In this paper, we show that a map satisfies for all rank r tensors if and only if is additive. Here r is a fixed integer such that and with , or with and . Examples showing the indispensability of assumption in our results are included.

A note on classical adjoint commuting linear maps of tensor products of matrices

Let be the algebra of matrices over a field with at least three elements. Inspired by linear preserver problems related to quantum information science, we characterize classical adjoint commuting linear maps, i.e. linear maps such that with , that preserve decomposable tensors of . Some examples are given to distinguish the study of classical adjoint commuting linear maps of tensor products of matrices from those found in the study of classical adjoint commuting linear maps on matrix spaces.

LINEAR SPACES AND PRESERVERS OF BOUNDED RANK-TWO PER-SYMMETRIC TRIANGULAR MATRICES

Let F be a field and m,n be integers m,n > 3. Let SMn(F) and STn(F) denote the linear space of n × n per-symmetric matrices over F and the linear space of n × n per-symmetric triangular matrices over F, respectively. In this note, the structure of spaces of bounded rank-two matrices of STn(F) is determined. Using this structural result, a classification of bounded rank-two linear preservers : STn(F) ! SMm(F), with F of characteristic not two, is obtained. As a corollary, a complete description of bounded rank-two linear preservers between per-symmetric triangular matrix spaces over a field of characteristic not two is addressed.

COMPOUND-COMMUTING ADDITIVE MAPS ON MATRIX SPACES

In this note, compound-commuting additive maps on matrix spaces are studied. We show that compound-commuting additive maps send rank one matrices to matrices of rank less than or equal to one. By using the structural results of rank-one nonincreasing additive maps, we characterize compound-commuting additive maps on four types of matrices: triangular matrices, square matrices, symmetric matrices and Hermitian matrices.