Majorisierungseigenschaften

Majorisierungseigenschaften, also known as majorization properties, are a fundamental concept in matrix theory and inequality analysis. The principle states that for two real vectors \(x = (x_1,\dots,x_n)\) and \(y = (y_1,\dots,y_n)\), sorted in non‑increasing order, \(x\) is said to majorize \(y\) (written \(x \succ y\)) if the partial sums satisfy \(\sum_{i=1}^k x_i \ge \sum_{i=1}^k y_i\) for all \(k = 1,\dots,n-1\) and the totals are equal, \(\sum_{i=1}^n x_i = \sum_{i=1}^n y_i\). This preorder on \(\mathbb R^n\) underpins various comparison results; in particular, a function that preserves majorization is called Schur‑convex, and one that reverses it is Schur‑concave. Many classical inequalities, such as the rearrangement inequality and the Hardy–Littlewood–Pólya inequality, are direct consequences of majorization. In linear algebra, the eigenvalue vectors of Hermitian matrices obey majorization relations when sums or products of matrices are considered, leading to bounds on spectral norms and trace functions. In probability theory, majorization provides a tool for comparing distributions through stochastic ordering. Applications extend to economics, via the theory of income inequality, and to quantum information, where it characterizes state transformations under local operations and classical communication. The study of majorization continues to be an active research area, linking combinatorics, optimization, and matrix analysis.