This paper considers the problem of constructing a normatively significant multidimensional Gini index of relative inequality. The social evaluation relation (SER) from which the index is derived is required to satisfy a weak version of the Pigou-Dalton Bundle Principle (WPDBP) (rather than Uniform Majorization or similar conditions). It is also desired to satisfy a weak form of the condition o...

We discuss the l2-lp (with p ∈ (0, 1)) matrix minimization for recovering low rank matrix. A smoothing approach is developed for solving this non-smooth, non-Lipschitz and non-convex optimization problem, in which the smoothing parameter is used as a variable and a majorization method is adopted to solve the smoothing problem. The convergence theorem shows that any accumulation point of the seq...

Horn’s Theorem plays an important role in the theory of matrix majorization and elsewhere. We give a simple proof of it, as well as a related theorem of Mirsky.

insisting that the equality sign holds when k = n. Here, x[X] > • • • > x[n] are the xt arranged in decreasing order and, similarly, y[X] > • • • > y[n]. If (1) is only required for the increasing (decreasing) convex functions on R then one speaks of weak sub-majorization x >, respectively). The first is equivalent to (2). Let & be an open convex subset of R...

The objective of this study is to develop a majorization-based tool to compare financial networks with a focus on the implications of liability concentration. Specifically, we quantify liability concentration by applying the majorization order to the liability matrix that captures the interconnectedness of banks in a financial network. We develop notions of balancing and unbalancing networks to...

In this expository study some basic matrix inequalities obtained by embedding bilinear forms 〈Ax, x〉 and 〈Ax, y〉 into 2 × 2 matrices are investigated. Many classical inequalities are reproved or refined by the proposed unified approach. Some inequalities involving the matrix absolute value |A| are given. A new proof of Ky Fan’s singular value majorization theorem is presented.

In this paper, by using majorization inequalities, upper bounds on summations of eigenvalues (including the trace) of the solution for the Lyapunov matrix differential equation are obtained. In the limiting cases, the results reduce to bounds of the algebraic Lyapunov matrix equation. The effectiveness of the results are illustrated by numerical examples.