Finally, in section 4, we show a role of gdoubly stochastic operators in the theory. Theory of majorization and its applications 1980 albert w. Linear matrix inequalities in system and control theory society for industrial and applied mathematics. Log majorization and complementary goldenthompson type. Majorization inequalities are useful and important. In this paper, we have extended some discrete majorization type inequalities of convex functions from intervals to rectangles. These keywords were added by machine and not by the authors. Majorizations and inequalities in matrix theory core. Niezgoda g majorization inequalities and canonical forms of matrices orderings. Massart is a most welcome and complete account on the modern developments of concentration inequalities. We show that a wide variety of problems arising in system and control theory can be reduced to a handful of standard convex and quasiconvex optimization problems that involve matrix inequalities. Request pdf weak majorization inequalities and convex functions let f be a convex function defined on an interval i, 01 and a,bn. Inequalities theory of majorization and its a, length. Matrix inequalities, majorization, positive linear maps, schur products.
Inequalities theory of majorization and its a, author. Matrix algebra and its applications to statistics and. Theory of majorization and its applications matrix analysis 1996 rajendra bhatia, springer, isbn 9780387948461. Trace inequalities for matrix products and trace bounds for. An eigenvalue majorization inequality for positive.
These bounds improve and extend the previous results. In this paper, we are interested in inequalities related to noninteger powers of generalized matrix functions. The given results are generalizations of the previously known results. The condition generalizes some known inequalities in the matrix theory involving eigenvalues or singular values of a matrix. Correspondingly, different types of matrix operator inequalities have become a big research field with wide applications. Request pdf g majorization inequalities for linear maps, ii in this article, we give a unified approach to vector inequalities of the form, where l and k are linear maps on a linear space. The corresponding majorization for acute triangles is identical to 8. H iai, log majorization and complementary goldenthompson type inequalities. For instance, majorization relations among eigenvalues and singular values of matrices produce a lot of norm inequalities and even matrix inequalities. By using diagonalizable matrix decomposition and majorization inequalities, we propose new trace bounds for the product of two real square matrices in which one is diagonalizable.
The basic topic of this book is solving problems from system and control theory using convex optimization. In matrix theory majorization also occurs in the comparison of eigenvalues with singular values 24, 11 and in the study of incidence matrices 21. Matrix inequalities majorization theory can also be used to prove some wellknown matrix inequalities. Majorization is a basic concept in matrix theory, and has recently become a useful mathematical tool in quantum information theory, beginning with work of nielsen 1999 that linked it with quantum operations described. Convex functions and some classical inequalities pages 443480 download pdf. See l, 5, ii for theory of majorization for matrices.
Indeed,manymajorization concepts had been reinvented and often rechristened in dif. Schurs inequality let q be an n x n hermitian matrix. Theory of majorization and its applications issn book. Theory of majorization and its applications sciencedirect. We write q diagq, with equality when q is diagonal. Furthermore, we give some trace bounds for the solution of the algebraic riccati equations, which improve some of. Nov 15, 2014 read some majorization inequalities in euclidean jordan algebras, linear algebra and its applications on deepdyve, the largest online rental service for scholarly research with thousands of academic publications available at your fingertips. Majorization theory for optimization in wireless communications. Marshall, ingram olkin, academic press, isbn 9780124737501.
Queries from many readers have motivated changes in the way that some topics are presented. Weak majorization inequalities and convex functions request pdf. Weak majorization inequalities and convex functions. G majorization inequalities for linear maps, ii request pdf.
Theory of majorization and its applications issn book 143. Theory of majorization and its applications marshall and olkin,1979many researchers were unaware of the rich bodyof literature related to majorization that was scattered in journals in a wide variety of. Discrete majorization type inequalities for convex. Niezgoda gmajorization inequalities and canonical forms of matrices orderings. In matrix theory, majorization plays a significant role. The notion of majorization has its roots in matrix theory and mathematical inequalities. A comprehensive survey on this topic can be found in 80. One is majorization inequalities, the other is the generalized wielandt inequality. Furthermore, we give some trace bounds for the solution of the algebraic riccati equations, which improve some of the previous results under certain conditions. A new chapter, chapter 10, on matrix inequalities, which presents a variety of inequalities on the eigenvalues and singular values of matrices and unitarily invariant norms. It generalizes the discrete form of jensens inequality, and generalizes in turn to the concept of schurconvex functions. The problem of the reduction of a group majorization induced by an infinite group to a finite group majorization is discussed. A corresponding result for a skew symmetric matrix is derived.
Matrix norm inequalities related to majorization and means for matrices are also. A tribute to marshall and olkins book inequalities. Use features like bookmarks, note taking and highlighting while reading inequalities. Electronic journal of linear algebra issn 1081 3810. Correspondingly, different types of matrixoperator inequalities have become a big research field with wide applications. Group majorization and schur type inequalities, linear. For instance, majorization relations among eigenvalues and singular values of matrices produce a lot of norm inequalities and even matrix. Combinatorics are discussedprimarilyin chapter7, butchapters2, 6, and section 5. Trace inequalities for matrix products and trace bounds. Matrix theory and matrix inequalities marvin marcus professor of computer science university of california, santa barbara henryk minc professor of mathematics university of california, santa barbara dover publications, inc. Majorization is a basic concept in matrix theory, and has recently become a useful mathematical tool in quantum information theory, beginning with work of nielsen 1999 that linked it with quantum operations described by local operations and classical communication.
Discrete majorization type inequalities for convex functions. Linear matrix inequalities in system and control theory. Topics covered include the theory of majorization, variational principles for eigenvalues, operator monotone and convex functions, and perturbation of matrix functions and matrix inequalities. For rightangled triangles, the majorization is identical to 8. Inequalities theory of majorization and its a by lupejensen. The concept of majorization in rn is wellknown in matrix theory. Matrix analysis by rajendra bhatia pdf download this book presents a substantial part of matrix analysis that is functional analytic in spirit. The authors present an extremely rich collection of inequalities in a remarkably coherent and unified approach.
Elementary geometric inequalities are found in chapter 8. It is innovative, coherent, well written and, most importantly, a pleasure to read. This book can be used as a textbook or a supplement for a linear algebra and matrix theory class or a seminar for senior undergraduate or graduate students. We shall extend classical majorization inequality for majorized tuples and establish weighted versions of majorization. The main result of this paper is the following reverse majorization inequality for a hermitian positive semide.
Inequalities in the analytic theory of polynomials we show how techniques in matrix theory and majorization can be used to derive inequalities relating the zeros and critical points of a polynomial. Li gave the best bound of the difference between eigenvalues of a realcomplex hermitian matrix and the matrix after removing offdiagonal blocks. We show how techniques in matrix theory and majorization. Theory of majorization and its applications second. Majorization inequalities is an interesting area of study, both from the theoretical and applied point of view. Theory of majorization and its applications issn book 143 kindle edition by olkin, ingram, marshall, albert w download it once and read it on your kindle device, pc, phones or tablets. For example, discussion of lidskiis eigenvalue majorization inequalities was moved from a section primarily devoted to singular value inequalities to the. Majorizations and inequalities in matrix theory sciencedirect. Majorization plays a role 267 8 geometric inequalities 269 a inequalities for the angles of atriangle 271 b inequalities for the sides of atriangle 276 c inequalities forthe exradii andaltitudes 282 d inequalities forthe sides, exradii, and medians. Matrix norm inequalities related to majorization and means for matrices are also discussed. Many of the majorization results for eigenvalues and singular values and their.
Angles, majorization, wielandt inequality and applications. The vector of eigenvalues q majorizes the vector of diagonal entries, diagq. As a consequence, several recent results in matrix analysis and inner product. Operators and matrices volume 7, number 3 20, 733737 doi. Gmajorization inequalities and canonical forms of matrices. Some majorization inequalities in euclidean jordan algebras. Contributions this paper is focused on the derivation of dataprocessing and majorization inequalities for fdivergences, and it applies these inequalities to information theory and statistics. Araki 2 used the log majorization method to extend the trace inequality of lieb and thirring. Matrix inequalities from a two variables functional. Matrix theory a useful recent survey is chapter 3 of horn and johnson 1986, on inequalities for singular values and also for norms. This greatly expanded new edition includes recent research on stochastic, multivariate and group majorization, lorenz order, and applications in physics and chemistry, in economics and political science, in matrix inequalities, and in probability and statistics. The authors present an extremely rich collection of inequalities in a remarkably coherent and unified. In this chapter we introduce the concept of majorization, present its basic properties, and show a variety of matrix inequalities in majorization.
A wellbalanced approach to discussing the mathematical theory and applications to problems in other areas is an attractive feature of the book. In some situations, such inequalities can be obtained by using the theory of majorization and schur convex functions. In mathematics, karamatas inequality, named after jovan karamata, also known as the majorization inequality, is a theorem in elementary algebra for convex and concave realvalued functions, defined on an interval of the real line. Hardylittlewoodpolyas inequality and a new concept of. A corresponding result for a skew symmetric matrix. Read some majorization inequalities in euclidean jordan algebras, linear algebra and its applications on deepdyve, the largest online rental service for scholarly research with thousands of academic publications available at your fingertips. Some majorization inequalities in euclidean jordan. Majorization is an important tool in deriving matrix inequalities of eigenvalues, singular values, and matrix norms.
Majorization theorem for convexi able functions muhammad adil khan1, 1 department of mathematics, university of peshawar, peshawar, pakistan received march 16, 2011. A necessary and sufficient condition for a possibility of the reduction is given. Theory of majorization and its applications merits strong praise. The contributions of ky fan to matrix theory and operator theory are substan. Analytic inequalities are discussed in chapter 3 and in sections 16. Combinatorial analysis, geometric inequalities, matrix theory, and. It can be used as a textbook in courses on matrix algebra for statisticians, econometricians and mathematicians as well. Matrix theory is found especially in chapters 9 and 10, but also in. These inequalities stengthen known results such as the gausslucas theorem and mahlers inequality. A comprehensive survey on this topic can be found in 79. A publication of the international linear algebra society.
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