Linear Preserves of Logarithm Majorization

Let X, Y ∈ R n , X, Y 0, we say X logarithm majorized by Y , written X ≺log Y if log X ≺ log Y . Let M+ nm be the collection of matrices with positive entries. For X, Y ∈ M+ nm, it is said that X is logarithm column (row) majorized by Y , and is denoted as X ≺ column log Y (X ≺ row log Y ), if Xj ≺log Yj (Xi ≺log Yi) for all j = 1, 2, · · · m(i = 1, 2, · · · n), where Xj and Yj (Xi and Yi) are the ith column (row) of X and Y respectively. In the present paper, the relations column (row) logarithm majorization on M+ nm are studied and also all linear operators T : M+ nm −→ M+ nm preserving column (row) logarithm majorization will be characterized.