L-rays of permutation matrices and doubly stochastic matrices

Let MnMn be the space of real n×nn×n matrices. We investigate a linear transformation σ:Mn→Rnσ:Mn→Rn, called an L-ray (motivated by X-ray), which is defined in terms of sums of the entries in the blocks of a certain “L  -shaped” partition of the positions of a matrix A∈MnA∈Mn. We find descriptions of the image of the classes of permutation matrices and doubly stochastic matrices under this map, and show connections to majorization theory.