In each case equality holds if and only if the rows and are proportional. For a real or complex matrix a the inequality a 4 0 means that a is hermitian and negative semide. More accurate young, heinz, and holder inequalities for matrices. University of california berkeley, linear algebra qualifying exam problem. Well also introduce a software package called cvx, which you can use to solve semide. Request pdf holder type inequalities for matrices we discuss h. This result is a counterpart to the concavity property of matrix weighted geometric means. The weighted geometric mean of positive definite matrices shares similar properties. Proving that the pnorm is a norm is a little tricky and not particularly relevant to this course. Braatz large scale systems research laboratory, department of chemical engineering, university of illinois at urbanachampaign. Matrix holdermccarthy inequality via matrix geometric means. Known inequalities that everyone must know before going to a math contest. 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.
The main tool used is a holder type inequality for multiple integrals of functions which are applied to variables satisfying linear dependency relations. It is considered to be one of the most important inequalities in all of mathematics. To prove the triangle inequality requires the following classical result. Finally, we give the refinement and the reverse of the matrix holder inequality which holds for every unitarily invariant norm. One is the so called tracial matrix holder inequality. Holders inequality was first found by leonard james rogers rogers 1888, and discovered independently by holder 1889.
Youll need to use this for some homework and exam problems. An inequalityfor the trace ofmatrix products,usingabsolute. Solving linear inequalities inequalities are useful for comparing things that are not equal. This new inequality is useful for attacking and solving a variety of. As applications, we prove radons, jensens reverse and weighted power mean inequalities and their discrete versions. Article pdf available in mathematical proceedings of the royal irish. The chart below lists the symbols used in inequalities, their verbal equivalent, and an example where x represents an unknown value. A visual proof that p ab inequality of the ranks of matrices. Notes on vector and matrix norms university of texas at austin. On development of inverses of the cauchy and holder inequalities. Also we give counterexamples for expected holder type. Matrix or trace inequalities of holder type as well as weak.
Amini may 9, 2018 these notes provide a short introduction to common matrix norms. For minkowskis inequality is called the triangle inequality. A remainder term for h\ olders inequality for matrices and quantum. Holder type inequalities for matrices request pdf researchgate. Department of mathematics, faculty of chemical engineering technical university budapest sztoczek u.
Inequalities, vector and matrix norms, functional analysis, plasticity. 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. Social inequality is at the heart of the 2030 agenda for sustainable development. Pdf the case of equality in holders inequality for. Inequality regarding ranks of matrices problems in mathematics. 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. If a is real then this amounts to a at and a is then called symmetric. For latin america the worlds most unequal region social inequality also represents a structural feature and therefore a fundamental challenge. We establish a new reverse holder integral inequality and its discrete version. Matrix or trace inequalities of holder type as well as weak majorizations of. This linear matrix inequality specifies a convex constraint on y. Clearly, the 1norm and 2 norms are special cases of the pnorm.
Holders reverse inequality and its applications changjian zhao and wing sum cheung abstract. Holders inequality for matrices publications of the ias. Existence of three positive solutions for mpoint boundaryvalue problem with onedimensional plaplacian feng, hanying and ge, weigao, taiwanese journal of mathematics, 2010. 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. 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. 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.
A difference counterpart to a matrix holder inequality. Minkowskis inequality can be generalized in various ways also called minkowski inequalities. Yang department of mechanical engineering and applied mechanics, university of michigan, ann arbor, mi 48109, u. 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. Minkowski inequality, determinantal inequality, positive definite matrix. Pdf the case of equality in holders inequality for matrices and. Holders inequality is used to prove the minkowski inequality, which is the triangle inequality in the space l p.
An inequality for the trace of matrix products, using absolute values. It would be interesting to know if the inequality 2. This paper presents a survey of the development of inverses of the cauchy and holder inequalities, an approach to establishing bounds for the ratio of weighted means, a proof of equivalence of quotient inequality of weighted means and inverses of the generalized holder inequality, and extensions of the lyapunov inequality. When does the equality hold in the holder inequality. Serre, matrices, graduate texts in mathematics 216, 127. Matrix or trace inequalities of holder type as well as weak majorizations of similar type are. The case of equality in holders inequality for matrices and operators. The cauchyschwarz version involves the usual matrix. Actually, if we define the hermitian inner product. Determinant inequalities for hadamard product of positive.
It extends a scalar inequality due to gheorghiu and contains several kantorovich type inequalities. Review a tutorial on linear and bilinear matrix inequalities. In this handout, we will focus on solving linear inequalities. Recap 3 good ways to prove a functional inequality. Next, we take a closer look at each of the inequalities shown on the chart. There are at least two generalizations of holder inequality to the noncommutative case. 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.