Combinatorial Group Theory: COMBINATORIAL GROUP THEORY

Linear preserver problems: A brief introduction and some special techniques

Linear preservers on triangular matrices

Given an n × n real matrix A. the Ky Fan k-norm of it is defined as the sum of its k largest singular values. We determine the structures of the linear transformations T on that preserve the Ky Fan k-norm. We show that except for (n,k) = (4,2), there exist orthogonal matrices U and V such that where A+ either denotes A or denotes At . For the particular case (n,k) = (4, 2)T is either of the usual form mentioned above or equivalent to the composition of the usual form with a particular operator that has many remarkable properties.

Linear transformations on that preserve the Ky Fan k-norm and a remarkable special case when (n k) = (4, 2)

We prove the following result. Let F be an infinite field of characteristic other than two. Let k be a positive integer. Let Sn (F) denote the space of all n × n symmetric matrices with entries in F, and let T:Sn (F)→Sn (F) be a linear operator. Suppose that T is rank-k nonincreasing and its image contains a matrix with rank higher than K. Then, there exist λ∊F and P∊Fn,n such that T(A)=λPAPt for all A∊Sn (F). λ can be chosen to be 1 if F is algebraically closed and ±1 if F=R, the real field.

Linear mappings which are rank-k nonincreasing, II

Let U⊗V denote the tensor product of two finite dimensional vector spaces U and V over an infinite field. Let k be a positive integer such that kdim U and k dim V Let Dk denote the set of all non-zero elements of U⊗V of rank less than k. In this paper we study linear transformations T on U⊗V such that (TDk )⊆Dk .

G. H. Chan

Papers

Linear transformations on tensor spaces

Minimal (k)-groups of degree n 3k<n≦4k