Explore the vast range of titles available.

Intensive research in matrix completions, moments, and sums of Hermitian squares has yielded a multitude of results in recent decades. This book provides a comprehensive account of this quickly developing area of mathematics and applications and gives complete proofs of many recently solved problems. With MATLAB codes and more than 200 exercises, the book is ideal for a special topics course for graduate or advanced undergraduate students in mathematics or engineering, and will also be a valuable resource for researchers. Often driven by questions from signal processing, control theory, and quantum information, the subject of this book has inspired mathematicians from many subdisciplines, including linear algebra, operator theory, measure theory, and complex function theory. In turn, the applications are being pursued by researchers in areas such as electrical engineering, computer science, and physics. The book is self-contained, has many examples, and for the most part requires only a basic background in undergraduate mathematics, primarily linear algebra and some complex analysis. The book also includes an extensive discussion of the literature, with close to 600 references from books and journals from a wide variety of disciplines.

The positive semidefinite zero forcing number of a graph is a parameter that is important in the study of minimum rank problems. In this paper, we focus on the algorithmic aspects of computing this parameter. We prove that it is NP-complete to find the positive semidefinite zero forcing number of a given graph, and this problem remains NP-complete even for graphs with maximum vertex degree 7. We present a linear time algorithm for computing the positive semidefinite zero forcing number of generalized series–parallel graphs. We introduce the constrained tree cover number and apply it to improve lower bounds for positive semidefinite zero forcing. We also give formulas for the constrained tree cover number and the tree cover number on graphs with special structures.

Several important properties of positive semidefinite processes of Ornstein—Uhlenbeck type are analysed. It is shown that linear operators of the form $X\\mapsto AX+XA^{\\text{T}}$ with $A\\in M_{d}({\\Bbb R})$ are the only ones that can be used in the definition provided one demands a natural non-degeneracy condition. Furthermore; we analyse the absolute continuity properties of the stationary distribution (especially when the driving matrix subordinator is the quadratic variation of a d-dimensional Lévy process) and study the question of how to choose the driving matrix subordinator in order to obtain a given stationary distribution. Finally, we present results on the first and second order moment structure of matrix subordinators, which is closely related to the moment structure of positive semidefinite Ornstein—Uhlenbeck type processes. The latter results are important for method of moments based estimation.

In this paper, we give some nonnegative function inequalities for positive semidefinite matrices.

In this article, we obtain another proof of the following classical trace inequality which says that if A is a positive semidefinite matrix and B is a Hermitian matrix, then tr A α BA β B ≤ tr A γ BA δ B for all non-negative real numbers α, β, γ, δ for which α + β = γ + δ and max {α, β} ≤ max {γ, δ}. This is a generalization of trace inequalities due to T. Ando, F. Hiai, and K. Okubo for the special cases when γ = α + β, δ = 0 and when α = β = γ + δ 2 , namely tr ( A α + β 2 B ) 2 ≤ tr   A α B A β B ≤ tr   A α + β B 2 .

This paper focused on the generalized Bott-Duffin (GBD) inverse and the GBD elements in Banach algebra with involution and C∗ -algebra, as well as on the property of the p -positive semidefinite elements that are a generalization of the L -positive semidefinite matrices closely related to the GBD inverse. Also, using matrix equalities, inclusion relations of subspaces, and projectors, we established various characterizations of the GBD property in the matrix sets, especially on the set of L -positive semidefinite matrices. Additionally, we compared the methods and tools that we have at our disposal in the matrix set on one side and in Banach and C∗ -algebras on the other. Using the GBD inverse as an example, we would like to compare the results and their proofs in both sets and explain steps to quite easily skip from one set to the other, as well as situations in which we must pay additional attention in order to avoid mistakes.

In this note, we give several singular value inequalities involving convex and concave functions, which can be considered as generalizations of Al-Natoor et al.’s results (J. Math. Inequal. 17:581–589, 2023 ). Moreover, some of our results are the generalizations of Al-Natoor et al.’s inequalities (Adv. Oper. Theory 9:21, 2024 ).