Vector majorization, matrix majorizations and their applications

Vector majorization is a preorder notion on the set of real vectors. A vector x is majorized by a vector y if, for any k, the sum of k largest coordinates of x is not less that of y and the total sums must coincide. One of the earliest origins of majorization is in Economics, in comparisons of income inequality. For example, vector majorization can be expressed in terms of Lorenz curves and Dalton's principle of transfers.

There are many ways to generalize vector majorization to real matrices. A natural generalization is the strong majorization. A matrix A is strongly majorized by a matrix B if A = DB for some doubly stochastic square matrix D. Different types of matrix majorization have been applied to different areas of mathematics and applications. For example, directional majorization, another important generalization of the vector case, can be expressed in terms of Lorenz zonotopes, while row-stochastic majorization plays an important role in the theory of statistical experiments.

We will discuss majorization for matrix classes, in particular, the problem of finding minimal cover classes. We also characterize linear operators preserving and converting majorizations. This approach allows to apply methods specific to one type of majorization when investigating other majorizaitons. Finally, we provide characterizations of matrix majorizations for (0, 1) and (0, ±1)-matrices. These types of restrictions lead to a wide range of combinatorial results and problems.

Joint work with Geir Dahl and Alexander Guterman