G-Majorization Inequalities and Canonical Forms of Matrices

An Eaton system is connected with a decomposition statement for vectors of a linear space and with a scalar inequality related to the decomposition. The Singular Value Decomposition for the space of complex matrices associated with von Neumann's trace inequality is a typical example. We present a G-majorization inequality involving two orthoprojectors related to an Eaton system. The inequality generalizes a variety of majorization results on eigenvalues and singular values of matrices. A relationship between the inequality and canonical form theorems for certain spaces of matrices is shown. G-doubly stochastic operators are discussed.