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2013 | Buch

Combinatorial Matrix Theory and Generalized Inverses of Matrices

herausgegeben von: Ravindra B. Bapat, Steve J. Kirkland, K. Manjunatha Prasad, Simo Puntanen

Verlag: Springer India

Enthalten in: Springer Professional "Wirtschaft+Technik" , Springer Professional "Technik" , Springer Professional "Wirtschaft"

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Über dieses Buch

This book consists of eighteen articles in the area of `Combinatorial Matrix Theory' and `Generalized Inverses of Matrices'. Original research and expository articles presented in this publication are written by leading Mathematicians and Statisticians working in these areas. The articles contained herein are on the following general topics: `matrices in graph theory', `generalized inverses of matrices', `matrix methods in statistics' and `magic squares'. In the area of matrices and graphs, speci_c topics addressed in this volume include energy of graphs, q-analog, immanants of matrices and graph realization of product of adjacency matrices. Topics in the book from `Matrix Methods in Statistics' are, for example, the analysis of BLUE via eigenvalues of covariance matrix, copulas, error orthogonal model, and orthogonal projectors in the linear regression models. Moore-Penrose inverse of perturbed operators, reverse order law in the case of inde_nite inner product space, approximation numbers, condition numbers, idempotent matrices, semiring of nonnegative matrices, regular matrices over incline and partial order of matrices are the topics addressed under the area of theory of generalized inverses. In addition to the above traditional topics and a report on CMTGIM 2012 as an appendix, we have an article on old magic squares from India.

Inhaltsverzeichnis

Frontmatter

Skew Spectrum of the Cartesian Product of an Oriented Graph with an Oriented Hypercube

Abstract

Let σ be an orientation of a simple graph H yielding an oriented graph H ^σ. We define an orientation ψ to the Cartesian product G=H□Q _d of H with the hypercube Q _d by orienting the edges of G in a specific way. The skew adjacency matrices S(G ^ψ) obtained in this way for some special families of G answer some special cases of the Inverse Eigenvalue Problem. Further we present a new orientation ϕ to the hypercube Q _d for which the skew energy equals the energy of the underlying hypercube, distinct from the two orientations of hypercubes defined by Tian (Linear Algebra Appl. 435:2140–2149, 2011) and show how one of the two orientations of Q _d described by Tian is a special case of our method.

A. Anuradha, R. Balakrishnan

Notes on Explicit Block Diagonalization

Abstract

In these expository notes we present a unified approach to explicit block diagonalization of the commutant of the symmetric group action on the Boolean algebra and of the nonbinary and q-analogs of this commutant.

Murali K. Srinivasan

The Third Immanant of q-Laplacian Matrices of Trees and Laplacians of Regular Graphs

Abstract

Let A=(a _i,j)_1≤i,j≤n be an n×n matrix where n≥3. Let \(\operatorname{det2}(A)\) and \(\operatorname{det3}(A)\) be its second and third immanants corresponding to the partitions λ ₂=2,1ⁿ⁻² and λ ₃=3,1ⁿ⁻³, respectively. We give explicit formulae for \(\operatorname{det2}(A)\) and \(\operatorname{det3}(A)\) when A is the q-analogue of the Laplacian of a tree T on n vertices and when A is the Laplacian of a connected r-regular graph G.

R. B. Bapat, Sivaramakrishnan Sivasubramanian

Matrix Product of Graphs

Abstract

In this paper, we characterize the graphs G and H for which the product of the adjacency matrices A(G)A(H) is graphical. We continue to define matrix product of two graphs and study a few properties of the same product. Further, we consider the case of regular graphs to study the graphical property of the product of adjacency matrices.

K. Manjunatha Prasad, G. Sudhakara, H. S. Sujatha, M. Vinay

Determinant of the Laplacian Matrix of a Weighted Directed Graph

Abstract

The notion of weighted directed graph is a generalization of mixed graphs. In this article a formula for the determinant of the Laplacian matrix of a weighted directed graph is obtained. It is a generalization of the formula for the determinant of the Laplacian matrix of a mixed graph obtained by Bapat et al. (Linear Multilinear Algebra 46:299–312, 1999).

Debajit Kalita

From Multivariate Skewed Distributions to Copulas

Abstract

In this paper, a methodology is presented for constructing skewed multivariate copulas to model data with possibly different marginal distributions. Multivariate skew elliptical distributions are transformed into corresponding copulas in the similar way as the Gaussian copula and the multivariate t-copula are constructed. Three-parameter skew elliptical distributions are under consideration. For parameter estimation of the skewed distributions, the method of moments is used. To transform mixed third-order moments into a parameter vector, the star product of matrices is used; for star product and its applications, see, for example, Kollo (J. Multivar. Anal. 99:2328–2338, 2008) or Visk (Commun. Stat. 38:461–470, 2009). Results of the first applications are shortly described and referred to.

Tõnu Kollo, Anne Selart, Helle Visk

Revisiting the BLUE in a Linear Model via Proper Eigenvectors

Abstract

We consider two linear models,

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and

https://static-content.springer.com/image/chp%3A10.1007%2F978-81-322-1053-5_7/311152_1_En_7_IEq2_HTML.gif

, having different covariance matrices. Our main interest lies in question whether a particular given blue under

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continues to be a blue under

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. We give a thorough proof of a result originally due to Mitra and Moore (Sankhyā, Ser. A 35:139–152, 1973). While doing this, we will review some useful properties of the proper eigenvalues in the spirit of Rao and Mitra (Generalized Inverse of Matrices and Its Applications, 1971).

Jan Hauke, Augustyn Markiewicz, Simo Puntanen

Inference in Error Orthogonal Models

Abstract

Error Orthogonal Models constitute a very interesting class of models very useful in the design of experiments. The use of commutative Jordan algebras of symmetric matrices is used in order to perform statistical inference. The concept of segregation is introduced thus allowing the estimation of variance components.

Francisco Carvalho, João Tiago Mexia

On the Entries of Orthogonal Projection Matrices

Abstract

The present paper is concerned with characterizing entries of orthogonal projectors (i.e., a Hermitian idempotent matrices). On the one hand, several bounds for the values of the entries are identified. On the other hand, particular attention is paid to the question of how an orthogonal projector changes when its entries are modified. The modifications considered are those of a single entry and of an entire row or column. Some applications of the results in the linear regression model are pointed out as well.

Oskar Maria Baksalary, Götz Trenkler

Moore–Penrose Inverse of Perturbed Operators on Hilbert Spaces

Abstract

Rank-one perturbations of closed range bounded linear operators on Hilbert space are considered. The Moore–Penrose inverses of these operators are obtained. The results are generalized to obtain the Moore–Penrose inverse of operators of the form \(A+V_{1}GV_{2}^{*}\). Applications to nonnegativity of the Moore–Penrose inverse and operator partial orders are considered.

Shani Jose, K. C. Sivakumar

The Reverse Order Law in Indefinite Inner Product Spaces

Abstract

The aim of this short note is to present a few reverse order laws for the Moore–Penrose inverse and the group inverse (when it exists) in indefinite inner product spaces, with respect to the indefinite matrix product. We also point out its relationship with the star and sharp orders, respectively.

Sachindranath Jayaraman

Generalized Inverses and Approximation Numbers

Abstract

We derive estimates for approximation numbers of bounded linear operators between normed linear spaces. As special cases of our general results, approximation numbers of some weighted shift operators on ℓ ^p and those of isometries and projections of norm 1 are found. In the case of finite-rank operators, we obtain estimates for the smallest nonzero approximation number in terms of their generalized inverses. Also, we prove some results regarding the relation between approximation numbers and the closedness of the range of an operator. We recall that the closedness of the range is a necessary condition for the boundedness of a generalized inverse. We give examples illustrating the results and also show that certain inequalities need not hold.

K. P. Deepesh, S. H. Kulkarni, M. T. Nair

On the Level-2 Condition Number for Moore–Penrose Inverse in Hilbert Space

Abstract

We prove that \({\rm{cond}}_{\dagger}(T)-1\leq {\rm{cond}}^{[2]}_{\dagger}(T)\leq{\rm{cond}}_{\dagger}(T)+1\), where T is a linear operator in a Hilbert space, \({\rm{cond}}_{\dagger}(T)\) is the condition number of computing its Moore–Penrose inverse, and \({\rm{cond}}^{[2]}_{\dagger}(T)\) is the level-2 condition number of this problem.

Huaian Diao, Yimin Wei

Products and Sums of Idempotent Matrices over Principal Ideal Domains

Abstract

Writing a square matrix as a product of idempotent matrices attracted the attention of several linear algebraists. Equally interesting is the problem of writing a square matrix as a sum of idempotent matrices. Much work was done for real matrices and for matrices over other algebraic structures. We shall consider some of this work and present some new results for matrices over projective free rings.

K. P. S. Bhaskara Rao

Perfect Semiring of Nonnegative Matrices

Abstract

In this paper, it is shown that the semiring of nonnegative matrices satisfies descending chain condition on right and left ideals, i.e., it is left or right perfect if and only if it is closed under Drazin inverse of all elements. Furthermore, each nonnil right and left ideal contains a nonzero idempotent. This generalizes the known result on the characterization of finite semigroups of nonnegative matrices.

Adel Alahmedi, Yousef Alkhamees, S. K. Jain

Regular Matrices over an Incline

Abstract

We discuss the invertibility of incline matrices over DL, the set of all idempotent elements in an incline, and for matrices over an integral incline. We discuss the regularity of matrices over DL. We obtain equivalent conditions for the existence of various generalized inverses of an incline matrix. We provide an algorithm for the regularity of matrices over DL and illustrate with suitable examples.

AR. Meenakshi

Matrix Partial Orders Associated with Space Preorder

Abstract

In this expository article, we discuss some fundamentals of well-known matrix partial orders that are closely associated with space preorder on rectangular matrices. Particularly, we consider partial order defined by space decomposition, star ordering, and minus partial order for our discussion. These relations are closely associated with comparison of column spaces and row spaces of matrices. Results associated with selected matrix relation that are known in the literature along with some interesting observations are put together. At many places, though the proofs of several results are known in the past literature, by part or completely, for better reading purpose, independent proofs are provided.

K. Manjunatha Prasad, K. S. Mohana, Y. Santhi Sheela

An Illustrated Introduction to Some Old Magic Squares from India

Abstract

In this article we consider old magic squares from India associated with

Daivajna Varāhamihira (505–587 AD) and his Bṛhat Saṁhitā [39]: magic perfume;

Khajuraho 945 AD: Sir Alexander Cunningham (1814–1893) [14];

Dudhai (Jhansi district) early 11th century: Harold Hargreaves (b. 1876) [27];

Ṭhakkura Pherū (fl. 1291–1323): Gaṇitasārakaumudī: The Moonlight of the Essence of Mathematics [1];

Simon de la Loubère (1642–1729): Monsieur Vincent, Surat [3,15];

Major-General Robert Shortrede (1800–1868) [16], Gwalior 1483 [11, (1842)]; Andrew Hollingworth Frost (1819–1907) [23], Nasik [17, (1877)];

Nārāyaṇa Paṇḍita (fl. 1340–1400): Gaṇita Kaumudī [2, (1356)];

Srinivasa Aiyangar Ramanujan (1887–1920) [34,35,40,43]; Prasantha Chandra Mahalanobis (1893–1972).

Magic squares were once part of occult philosophy, but more recently, however, they form part of recreational mathematics. For the past 50 years or so, they have been studied in a matrix-theoretic setting. Our main interest is in the history and philosophy of magic squares and the related magic matrices and in the related bibliography and biographies. We try to illustrate our findings as much as possible and, whenever feasible, with images of postage stamps and other philatelic items.

George P. H. Styan, Ka Lok Chu

A Report on CMTGIM 2012, Manipal

International Workshop and Conference on Combinatorial Matrix Theory and Generalized Inverses of Matrices 02–07 & 10–11 January 2012 Department of Statistics, Manipal University, Manipal, India

R. B. Bapat, K. Manjunatha Prasad

Backmatter

Titel: Combinatorial Matrix Theory and Generalized Inverses of Matrices
herausgegeben von: Ravindra B. Bapat
Steve J. Kirkland
K. Manjunatha Prasad
Simo Puntanen
Verlag: Springer India
Electronic ISBN: 978-81-322-1053-5
Print ISBN: 978-81-322-1052-8
DOI: https://doi.org/10.1007/978-81-322-1053-5