Holders inequality was first found by leonard james rogers rogers 1888, and discovered independently by holder 1889. Article pdf available in mathematical proceedings of the royal irish. Braatz large scale systems research laboratory, department of chemical engineering, university of illinois at urbanachampaign. May 17, 2016 a read is counted each time someone views a publication summary such as the title, abstract, and list of authors, clicks on a figure, or views or downloads the fulltext. A holder type inequality for symmetric matrices 1245 suppose the nonnegative vector ug unwn is not a characteristic vector of s. In this paper, by virtue of an expression of matrix geometric means for positive semidefinite matrices via the moorepenrose inverse, we show matrix versions of the holdermccarthy inequality, the holder inequality and quasiarithmetic power means via matrix geometric means, and their reverses for positive definite matrices via the generalized kantorovich constant. An inequality for the trace of matrix products, using absolute values. Serre, matrices, graduate texts in mathematics 216, 127. Matrix holdermccarthy inequality via matrix geometric means. Matrix or trace inequalities of holder type as well as weak majorizations of. A difference counterpart to a matrix holder inequality. Introduction when establishing the convergence of the distribution of sums of functions of gaussian fields, by the method of moments, one is led to the study of the. Pdf the case of equality in holders inequality for.
Request pdf holder type inequalities for matrices we discuss h. As applications, we prove radons, jensens reverse and weighted power mean inequalities and their discrete versions. If a is real then this amounts to a at and a is then called symmetric. Clearly, the 1norm and 2 norms are special cases of the pnorm. A read is counted each time someone views a publication summary such as the title, abstract, and list of authors, clicks on a figure, or views or downloads the fulltext. Determinant inequalities for hadamard product of positive. Holders reverse inequality and its applications changjian zhao and wing sum cheung abstract. Matrix or trace inequalities of holder type as well as weak majorizations of similar type are. Pdf the case of equality in holders inequality for matrices and. University of california berkeley, linear algebra qualifying exam problem. Inequalities, vector and matrix norms, functional analysis, plasticity. Minkowski inequality, determinantal inequality, positive definite matrix. Known inequalities that everyone must know before going to a math contest. Inequality regarding ranks of matrices problems in mathematics.
Next, we take a closer look at each of the inequalities shown on the chart. Well also introduce a software package called cvx, which you can use to solve semide. Notes on vector and matrix norms university of texas at austin. When does the equality hold in the holder inequality. One is the so called tracial matrix holder inequality. Recap 3 good ways to prove a functional inequality. This result is a counterpart to the concavity property of matrix weighted geometric means. Existence of three positive solutions for mpoint boundaryvalue problem with onedimensional plaplacian feng, hanying and ge, weigao, taiwanese journal of mathematics, 2010. This linear matrix inequality specifies a convex constraint on y. This new inequality is useful for attacking and solving a variety of. It extends a scalar inequality due to gheorghiu and contains several kantorovich type inequalities.
In mathematics, the cauchyschwarz inequality, also known as the cauchybunyakovskyschwarz inequality, is a useful inequality encountered in many different settings, such as linear algebra, analysis, probability theory, vector algebra and other areas. Review a tutorial on linear and bilinear matrix inequalities. On development of inverses of the cauchy and holder inequalities. The relationships between the main classical inequalities can be represented in a tree diagram figure \1\, which shows that, for example, the cauchyschwarz inequality follows from holders inequality, etc. Also we give counterexamples for expected holder type.
It would be interesting to know if the inequality 2. Department of mathematics, faculty of chemical engineering technical university budapest sztoczek u. We establish a new reverse holder integral inequality and its discrete version. Actually, if we define the hermitian inner product. In this handout, we will focus on solving linear inequalities. A visual proof that p ab inequality of the ranks of matrices. Youll need to use this for some homework and exam problems. To prove the triangle inequality requires the following classical result. More accurate young, heinz, and holder inequalities for matrices. Finally, we give the refinement and the reverse of the matrix holder inequality which holds for every unitarily invariant norm.
Theorem 4 extends the classical minkowski determinant inequality to all of the coefficients in the characteristic polynomial and theorem 5 extends a recent concavity result of fan 1 for determinants. Minkowskis inequality can be generalized in various ways also called minkowski inequalities. The chart below lists the symbols used in inequalities, their verbal equivalent, and an example where x represents an unknown value. The weighted geometric mean of positive definite matrices shares similar properties. There are at least two generalizations of holder inequality to the noncommutative case. For latin america the worlds most unequal region social inequality also represents a structural feature and therefore a fundamental challenge. In each case equality holds if and only if the rows and are proportional. Solving linear inequalities inequalities are useful for comparing things that are not equal.
An inequalityfor the trace ofmatrix products,usingabsolute. A remainder term for h\ olders inequality for matrices and quantum. We only need to prove the ag inequality because the hg inequality follows from the ag inequality and properties of the means ha 1 a 1 a. Matrix or trace inequalities of holder type as well as weak. For minkowskis inequality is called the triangle inequality. The case of equality in holders inequality for matrices and operators. The main tool used is a holder type inequality for multiple integrals of functions which are applied to variables satisfying linear dependency relations. Yang department of mechanical engineering and applied mechanics, university of michigan, ann arbor, mi 48109, u. Social inequality is at the heart of the 2030 agenda for sustainable development. Holders inequality is used to prove the minkowski inequality, which is the triangle inequality in the space l p. Amini may 9, 2018 these notes provide a short introduction to common matrix norms. It is considered to be one of the most important inequalities in all of mathematics. The cauchyschwarz version involves the usual matrix.
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