Involving Hadamard Products Research Articles

Two inequalities involving Hadamard products of positive semi-definite Hermitian matrices

We extend two inequalities involving Hadamard products of positive definite Hermitian matrices to positive semi-definite Hermitian matrices. Simultaneously, we also show the sufficient conditions for equalities to hold. Moreover, some other matrix inequalities are also obtained. Our results and methods are different from those which are obtained by S. Liu in [J. Math. Anal. Appl. 243:458-463(2000)] and B.-Y. Wang et al in [Lin. Alg. Appl. 302-303:163-172(1999)].

Several inequalities involving Khatri–Rao products of positive semidefinite matrices

The Khatri–Rao and Tracy–Singh products for partitioned matrices are viewed as generalized Hadamard and generalized Kronecker products, respectively. Based on the connection between the Khatri–Rao and Tracy–Singh products, we establish new inequalities involving Khatri–Rao products of positive semidefinite matrices. The results lead to inequalities involving Hadamard products, as a special case.

Inequalities Involving Hadamard Products of Positive Semidefinite Matrices

An inequality established by G. P. H. Styan (1973, Linear Algebra Appl.,6, 217–240) is on the Hadamard product and a correlation matrix. An inequality obtained by B.-Y. Wang and F. Zhang (1997, Linear and Multilinear Algebra, 43, 315–326) involves the Hadamard product and Schur complements. These two inequalities hold in the positive definite matrix case. Based on Albert's theorem, we present their extensions to cover the positive semidefinite matrix case. We also give relevant inequalities.

Extremal matrices in certain determinantal inequalities

The extremal matrices in certain inequalities for determinants of sums are characterized. Related determinantal inequalities involving Hadamard products of positive definite matrices are presented. These inequalities are easy consequences of majorization results recently obtained by Ando and Visick.

Spectral variations and Hadamard products: Some problems

Many perturbation problems in matrix theory are related to three matrix minimization problems involving Hadamard products. Possible solutions to the problems are conjectured. The problems, if solved, may imply significant advances in eigenvalue, generalized eigenvalue variations, and perhaps in other aspects in matrix theory, too.

Inequalities for the Hadamard Product of Matrices

Previous article Next article Inequalities for the Hadamard Product of MatricesB. Mond and J. E. PecaricB. Mond and J. E. Pecarichttps://doi.org/10.1137/S0895479896302953PDFBibTexSections ToolsAdd to favoritesExport CitationTrack CitationsEmail SectionsAboutAbstractSome known inequalities for the Hadamard product of matrices are extended and new inequalities obtained.[1] Roger Horn, The Hadamard product, Proc. Sympos. Appl. Math., Vol. 40, Amer. Math. Soc., Providence, RI, 1990, 87–169 1059485 Google Scholar[2] Google Scholar[3] Tōnu Kollo and , Heinz Neudecker, Asymptotics of eigenvalues and unit‐length eigenvectors of sample variance and correlation matrices, J. Multivariate Anal., 47 (1993), 283–300 10.1006/jmva.1993.1084 94j:62040 CrossrefISIGoogle Scholar[4] Shuangzhe Liu and , Heinz Neudecker, Several matrix Kantorovich‐type inequalities, J. Math. Anal. Appl., 197 (1996), 23–26 10.1006/jmaa.1996.0003 97d:26023 CrossrefISIGoogle Scholar[5] B. Mond and , J. Pečarić, On Jensen’s inequality for operator convex functions, Houston J. Math., 21 (1995), 739–754 96m:47028 ISIGoogle Scholar[6] Daisuke Kainuma and , Masahiro Nakamura, Around Jensen’s inequality, Math. Japon., 25 (1980), 585–588 82d:46091 Google Scholar[7] A. Marshall and , I. Olkin, Matrix versions of the Cauchy and Kantorovich inequalities, Aequationes Math., 40 (1990), 89–93 91f:15045 CrossrefGoogle Scholar[8] B. Mond and , J. Pečarić, A matrix version of the Ky Fan generalization of the Kantorovich inequality. II, Linear and Multilinear Algebra, 38 (1995), 309–313 96h:15021 CrossrefGoogle ScholarKeywordsmatrix inequalitiesHadamard products Previous article Next article FiguresRelatedReferencesCited byDetails On the Restricted Isometry of the Columnwise Khatri–Rao ProductIEEE Transactions on Signal Processing, Vol. 66, No. 5 Cross Ref Several inequalities involving Khatri–Rao products of positive semidefinite matricesLinear Algebra and its Applications, Vol. 354, No. 1-3 Cross Ref Two inequalities involving Hadamard products of positive semi-definite Hermitian matricesJournal of Applied Mathematics and Computing, Vol. 10, No. 1-2 Cross Ref Inequalities involving Khatri-Rao products of Hermitian matricesKorean Journal of Computational & Applied Mathematics, Vol. 9, No. 1 Cross Ref Volume 19, Issue 1| 1998SIAM Journal on Matrix Analysis and Applications History Published online:31 July 2006 InformationCopyright © 1998 Society for Industrial and Applied MathematicsKeywordsmatrix inequalitiesHadamard productsMSC codes15A45PDF Download Article & Publication DataArticle DOI:10.1137/S0895479896302953Article page range:pp. 66-70ISSN (print):0895-4798ISSN (online):1095-7162Publisher:Society for Industrial and Applied Mathematics

The Hadamard Operator Norm of a Circulant and Applications

Let $M_n $ be the space of $n \times n$ complex matrices and let $\| \cdot \|_\infty $ denote the spectral norm. Given matrices $A = [ a_{ij} ]$ and $B = [ b_{ij} ]$ of the same size, define their Hadamard product to be $A \circ B = [ a_{ij} b_{ij} ]$. Define the Hadamard operator norm of $A \in M_n $ to be \[ \| | A | \|_\infty = \max \{ \| A \circ B \|_\infty : \| B \|_\infty \leq 1 \}. \] It is shown that \[ (1) \qquad |||A|||_\infty = {\operatorname{tr}} | A |/n \] if and only if \[ (2) \qquad | A | \circ I = | A^ * | \circ I = ( {\operatorname{tr}} | A |/n ) I. \] It is shown that (2) holds for generalized circulants and hence that the Hadamard operator norm of a generalized circulant can be computed easily. This allows us to compute or bound \[ |||{\operatorname{sign}} ( j - i )_{i,j = 1}^n |||_\infty ,\quad ||| [ ( \lambda _i - \lambda _j )/( \lambda _i + \lambda _j ) ]_{i, j = 1}^n |||_\infty, \quad |||T_n |||_\infty, \] where $T_n $ is the $n \times n$ matrix with ones on and above the diagonal and zeros below, and related quantities. In each case the norms grow like log n. Using these results upper and lower bounds are obtained on quantities of the form \[ \sup \{ \| \,| A | - | B |\, \|_\infty : \| A - B \|_\infty \leq 1,A,B \in M_n \} \] and \[ \sup \{ \| \, | A | B - B | A |\, \|_\infty : \| AB - BA \|_\infty \leq 1,A,B \in M_n , A = A^ * \}. \] The authors also indicate the extent to which the results generalize to all unitarily invariant norms, characterize the case of equality in a matrix Cauchy–Schwarz Inequality, and give a counterexample to a conjecture involving Hadamard products.

An arithmetic-geometric-harmonic mean inequality involving Hadamard products

Given matrices of the same size, A = [ a ij ] and B = [ b ij ], we define their Hadamard product to be A ∘ B = [ a ij b ij ]. We show that if x i > 0 and q ⩾ p ⩾ 0, then the n × n matrices x ix j x i+x j , , x i −1+x j −1 x ix j ,and x i p+x j p x i q+x j q are positive definite, and we relate these facts to some matrix valued arithmetic- geometric-harmonic mean inequalities—some of which involve Hadamard products and others unitarily invariant norms. It is known that if A is positive semidefinite then max{| A∘ B|:| B|⩽1} = max a ij , where |·| denotes the spectral norm. We show that the converse of this statement is false and give a useful partial converse.