Linear algebraic and combinatorial problems in matrix majorization

The origins of majorization theory can be found in a variety of different contexts. The most notable examples are Muirhead's work on symmetric polynomials, Schur's work on Hadamard's inequality, and works by Lorenz and Dalton on income distributions. Hardy, Littlewood, and Polya laid the foundations of majorization as an independent theory.
 
Vector majorization is a preorder notion on the real vector space. A vector x is majorized by a vector y if, for any k, the sum of the k largest coordinates of x is not less than the sum for y, and the total sums coincide. Later this notion was generalized to matrices and other structures. Matrix and vector majorizations have numerous applications in many areas of mathematics.
 
We will discuss the notion of majorization for matrix classes. Using a geometric approach, we solve the problem of finding minimal covering classes. We characterize majorizations for (0, 1) and (0, \pm 1)-matrices and discuss connections between these results and linear programming and graph theory. Finally, we will consider linear preserver problems connected to majorization. With the help of combinatorial matrix theory, we are able to prove the classical results of Ando and Li, Poon under weaker assumptions. In particular, we characterize linear operators preserving strong majorization for (0, 1)-matrices.
 
The talk is based on results of joint work with Geir Dahl and Alexander Guterman.