Completing block Hermitian matrices with maximal and minimal ranks and inertias

Tian, Yongge

doi:10.13001/1081-3810.1419

Cited by 9 publications

(6 citation statements)

References 36 publications

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“…where K is an example of a block diagonal partial Hermitian matrix (Tian 2010). Clearly, the number of individual elements to be estimated in K is far less than in G. For example, for n p 8,750 and k p 50, as in our empirical analysis below, K contains 223,125 unique elements, compared to 3.83#10 7 unique elements in G. Therefore, only 0.6% of the unique elements of G are contained in K. Such an enormous difference in the number of elements to be estimated provides a strong incentive to find ways to complete G when only K is estimated from the data (Candes and Recht 2009;Tian 2010).…”

Section: The Problem Of Matrix Completionmentioning

confidence: 99%

The Phenome-Wide Distribution of Genetic Variance

Blows¹,

Allen²,

Chenoweth³

et al. 2015

The American Naturalist

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JSTOR is a not-for-profit service that helps scholars, researchers, and students discover, use, and build upon a wide range of content in a trusted digital archive. We use information technology and tools to increase productivity and facilitate new forms of scholarship. For more information about JSTOR, please contact support@jstor.org. abstract: A general observation emerging from estimates of additive genetic variance in sets of functionally or developmentally related traits is that much of the genetic variance is restricted to few trait combinations as a consequence of genetic covariance among traits. While this biased distribution of genetic variance among functionally related traits is now well documented, how it translates to the broader phenome and therefore any trait combination under selection in a given environment is unknown. We show that 8,750 gene expression traits measured in adult male Drosophila serrata exhibit widespread genetic covariance among random sets of five traits, implying that pleiotropy is common. Ultimately, to understand the phenome-wide distribution of genetic variance, very large additive genetic variancecovariance matrices (G) are required to be estimated. We draw upon recent advances in matrix theory for completing high-dimensional matrices to estimate the 8,750-trait G and show that large numbers of gene expression traits genetically covary as a consequence of a single genetic factor. Using gene ontology term enrichment analysis, we show that the major axis of genetic variance among expression traits successfully identified genetic covariance among genes involved in multiple modes of transcriptional regulation. Our approach provides a practical empirical framework for the genetic analysis of highdimensional phenome-wide trait sets and for the investigation of the extent of high-dimensional genetic constraint.

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Section: The Problem Of Matrix Completionmentioning

confidence: 99%

The Phenome-Wide Distribution of Genetic Variance

Blows¹,

Allen²,

Chenoweth³

et al. 2015

The American Naturalist

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“…As direct applications, we gave necessary and sufficient conditions for the existence of X satisfying the triple matrix equations in (1.2) and (1.3), as well as some matrix inequalities. Although the problems of maximizing and minimizing ranks and inertias of matrices are generally regarded as NP-hard, the results presented in this previous sections as well as the papers [13,15,16,17,18,30,31,32,36,38] show that many closed-form formulas for calculating global extremal ranks inertias of some simpler matrix expressions can be established symbolically by using some pure algebraic operations of matrices, while these explicit formulas can be used to solve many fundamental problems in matrix theory, as mentioned in the beginning of this paper. All the results obtained in these papers are brand-new, but easy to understand within the scope of elementary linear algebra.…”

Section: Discussionmentioning

confidence: 99%

“…This basic work was also extended to some general LMFs, such as, A − BX − (BX) * , A − BXB * − CY C * and A − BXC − (BXC) * , where X and Y are (Hermitian) variable matrices of appropriate sizes; see [2,15,16,17,18,32,33,36]. We shall use some pure algebraic operations on matrices to derive two groups of analytical formulas for calculating the global extremal values of the objective functions in (1.4)-(1.7) and (1.9)- (1.12), and then to present a variety of valuable consequences of these formulas.…”

Section: Introductionmentioning

confidence: 99%

Some optimization problems on ranks and inertias of matrix-valued functions subject to linear matrix equation restrictions

Tian¹

2014

Banach J. Math. Anal.

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Matrix rank and inertia optimization problems are a class of discontinuous optimization problems in which the decision variables are matrices running over certain matrix sets, while the ranks and inertias of the variable matrices are taken as integer-valued objective functions. In this paper, we establish a group of explicit formulas for calculating the maximal and minimal values of the rank and inertia objective functions of the Hermitian matrix expression A1 − B1XB * 1 subject to the common Hermitian solution of a pair of consistent matrix equations B2XB * 2 = A2 and B3XB * 3 = A3, and Hermitian solution of the consistent matrix equation B4X = A4, respectively. Many consequences are obtained, in particular, necessary and sufficient conditions are established for the triple matrix equations B1XB * 1 = A1, B2XB * 2 = A2 and B3XB * 3 = A3 to have a common Hermitian solution, as necessary and sufficient conditions for the two matrix equations B1XB * 1 = A1 and B4X = A4 to have a common Hermitian solution. are given, i = 1, 2, 3, and X ∈ C n H is a variable matrix, and study the following constrained optimization problems: Problem 1.1 For the constrained linear matrix-valued function in (1.2), establish explicit formulas for calculating max X∈C n H r(

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“…(i) the minimum of {rk(Ã)|Ã completion of A} is Here we determine the maximum rank of the symmetric completions of a symmetric partial matrix where only the diagonal blocks are given (see Theorem 5) and the minimum rank and the maximum rank of the antisymmetric completions of an antisymmetric partial matrix where only the diagonal blocks are given (see Theorem 9). In [3], [4], [9], the analogous problem has been solved for hermitian matrices.…”

Section: Introductionmentioning

confidence: 99%