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In this note, we characterize linear maps between spaces of Hermitian matrices over a field with involution which preserve the rank of the identity matrix and the rank of tensor products of rank one Hermitian matrices. It is assumed that the fixed field with respect to the field with involution has at least four elements. We also present some applications of our main result to other preserver problems. [ABSTRACT FROM AUTHOR]

In this note, we give a simple proof as well as an extension of a very recent result of B. Zheng, J. Xu and A. Fosner concerning linear maps between vector spaces of complex square matrices that preserve the rank of tensor products of matrices by using a structure theorem of R. Westwick on linear maps between tensor product spaces that preserve non-zero decomposable elements. [ABSTRACT FROM AUTHOR]

Letdenote the two element Boolean algebra andbe the semimodule of alln-square symmetric Boolean matrices with zero diagonal. We characterize (i) subsemimodules ofwhose nonzero members all have factor rank 2, (ii) linear mappings fromtothat send distinct elements of term rank 2 to distinct elements of factor rank 2 and (iii) linear mappings fromtothat preserve elements of factor rank 2 and also elements of factor rankkfor some [ABSTRACT FROM PUBLISHER]

Abstract: Let be the semimodule of all Boolean matrices where is the Boolean algebra with two elements. Let k be a positive integer such that . Let denote the subsemimodule of spanned by the set of all rank k matrices. We show that if T is a bijective linear mapping on , then there exist permutation matrices P and Q such that for all or and for all . This result follows from a more general theorem we prove concerning the structure of linear mappings on that preserve both the weight of each matrix and rank one matrices of weight . Here the weight of a Boolean matrix is the number of its nonzero entries. [ABSTRACT FROM AUTHOR]