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.

Extracting structured subgraphs inside large graphs – often known as the planted subgraph problem – is a fundamental question that arises in a range of application domains. This problem is NP-hard in general, and as a result, significant efforts have been directed towards the development of tractable procedures that succeed on specific families of problem instances. We propose a new computation...

A sectorial matrix is an n×n whose numerical range contained in open half-plane, and such matrices have many nice properties. In particular, the subset of strictly accretive a convex cone space matrices, results related to positive definite recently been generalized this cone. Moreover, used define phases matrix, these can be angularly bound eigenvalues by majorization-type inequalities similar...

A lower bound on the chromatic number of a graph is derived by majorization of spectra of weighted adjacency matrices. These matrices are given by Hadamard products of the adjacency matrix and arbitrary Hermitian matrices.