M.H. Lim

Linear transformations on symmetric matrices that preserve commutativity

It is shown that if a nonsingular linear transformation T on the space of n-square real symmetric matrices preserves the commutativity, where n ⩾3, then T(A) = λQAQt + Q(A)In for all symmetric matricesA, for some scalar λ, orthogonal matrix Q, and linear functional Q.

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 ).

Linear transformations on symmetric matrices II

Let Tbe a linear mapping on the space of n× nsymmetric matrices over a field Fof characteristic not equal to two. We obtain the structure of Tfor the following cases:(i) Tpreserves matrices of rank less than three; (ii) Tpreserves nonzero matrices of rank less than K + 1 where Kis a fixed positive integer less than nand Fis algebraically closed; (iii) Tpreserves rank Kmatrices where Kis a fixed odd integer and Fis algebraically closed.

Linear mappings on second symmetric product spaces that Preserve rank less than or equal to one

In this paper we characterize those linear mappings from a second symmetric product space to another which preserve decomposable elements of the form λu[sdot]u where u is a vector and λ is a scalar. This leads to the corresponding result concerning linear mappings from one vector space of symmetric matrices to another which preserve rank less than or equal to one. We also discuss some consequences of this characterization theorem.

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.

Linear transformations on symmetry classes of tensors II

Let U be a vector space over a field of characteristic 0. We show in this paper that every surjective linear mapping from the mth tensor space of U to the mth symmetric product space over U that takes nonzero decomposable elements to nonzero decomposable elements is induced by m isomorphisms on U if dim U ≥ 3. The result is applied to symmetry classes of tensors over U associated with permutation groups and the character identically 1.

Linear transformation on symmetry classes of tensors

This paper is concerned with linear transformations on a symmetry class of tensors preserving nonzero decomposable elements.

A note on the relation between the determinant and the permanent

In this note we prove that there is no linear mapping T on the space of n-square symmetric matrices over any subfield of real field such that the determinant of A is equal to the permanent of T(A) for all symmetric matrices A if n3.

A note on similarity preserving linear maps on matrices

Abstract We give a simple proof of a result of Hiai concerning similarity preserving linear maps on n × n complex matrices as well as its extension to fields of characteristic zero.

Linear transformations on symmetric matrices that preserve the permanent

Abstract It is shown that if a linear transformation T on the space of n -square symmetric matrices over any subfield of the real field preserves the permanent, where n ⩾ 3, then T ( A )= ± PAP t for all symmetric matrices A and a fixed generalized permutation matrix P with per P = ± 1.