Some Trace Inequalities for Positive Semidefinite Matrices
Ramazan Türkmen *
Mathematics Department, Science Faculty, Selcuk University, 42130, Konya, Turkey.
İsaf Gamze Kurudere
Mathematics Department, Süleyman Demirel University, Institute of Science, 32260, Isparta, Turkey.
*Author to whom correspondence should be addressed.
Abstract
In this study, the general properties of positive semidefinite matrices are examined, and in the beginning of our work, the definition and basic properties of the matrix trace. Hermitian matrices and their fundamental properties, as well as positive semi definite matrices and their essential characteristics are presented. Trace inequalities involving positive semidefinite matrices play an essential role in many areas such as control theory, quantum information theory, and operator theory. Owing to these applications, the investigation of trace inequalities has attracted considerable attention in recent years from both theoretical and applied viewpoints. In this work, firstly, the basic theorems and proofs related to positive semidefinite matrices, trace inequalities of matrices, block matrices, Shur complements of block matrices and Hadamard products are studied. In the last part of the work, some new trace inequalities are found for traces of products and sums of positive semi definite block matrices.
Keywords: Block matrix, Hadamard product, Hermitian matrix, matrix inequalities, matrix trace, positive semidefinite matrix