保对称矩阵张量积秩的线性映射

Linear maps preserving rank of tensor products of symmetric matrices

设Sm是复数域■上m×m对称矩阵全体。线性映射φ:S■mSn→Smn保持矩阵张量积秩,即rankφ(A■B)=rank(A■B),■A∈Sm,B∈Sn当且仅当存在可逆阵P∈Mmn使得φ(X)=PXPt,■X∈S■mSn。本文是对矩阵张量积空间上的线性保持问题的补充和发展。

Let Sm be the space of all m×m symmetric matrices over a complex field.A linear mapφ:S■mSn→Smn is said to be a rank preserver if rankφ(A■B)=rank(A■B),■A∈Sm,B∈Sn.It is shown thatφ:S■mSn→Smn is a linear map preserving rank if and only if there exists an invertible matrix P∈Mmn such thatφ(X)=PXPt for every X∈S■mSn.This paper is a supplement and development of the linear preserving problem on the tensor product space of matrices.