IIT Madras is a public technical and research university located in Chennai, Tamil Nadu. Founded in 1959, it is recognised as an Institute of National Importance.

IIT Madras has been ranked as the top engineering institute in India for four years in a row by the National Institutional Ranking Framework of the MHRD

It currently offers undergraduate, postgraduate and research degrees across 16 disciplines in Engineering, Sciences, Humanities and Management. About 596 faculty belonging to science and engineering departments and centres of the Institute are engaged in teaching, research and industrial consultancy.

IIT Madras

The authors introduce a new type of matrix splitting generalizing the notion of B-splitting and study its relationships with nonnegativity of the Moore-Penrose inverse and the group inverse. © element, Zagreb.

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Department of Mathematics

Operators and Matrices

On splittings of matrices and nonnegative generalized inverses

For A,B∈Rm×n, let J=[A,B] be the set of all matrices C such that A≤C≤B, where the order is component wise. Krasnosel'skij et al. [9] and Rohn [11] have shown that if A and B are invertible with A- 1≥0 and B-1≥0, then every C∈J is invertible with C-1≥0. In this article, we present certain extensions of this result to the singular case, where the nonnegativity of the usual inverses is replaced by the nonnegativity of the Moore-Penrose inverse. © 2011 Elsevier Inc. All rights reserved.

Fulltext

Linear Algebra and Its Applications

Moore-Penrose inverse positivity of interval matrices

The authors revisit the notion of a row monotone matrix and obtain new results that establish relationships with certain specific types of splittings of the matrix concerned. Among other results, analogous results for range monotone matrices are also obtained. © 2011 Elsevier Inc. All rights reserved.

Generalizations of matrix monotonicity and their relationships with certain subclasses of proper splittings

A dominance rule for singular matrices using proper splittings is proposed. This extends the corresponding notion, known for nonsingular matrices. An application to the nonnegativity of the Moore-Penrose inverse is presented. © 2012 Copyright Taylor and Francis Group, LLC.

Linear and Multilinear Algebra

A dominance notion for singular matrices with applications to nonnegative generalized inverses

In this article, a classical result of Cottle and Veinott characterizing least elements of polyhedral sets in terms of nonnegative left-inverses is extended to characterize nonnegativity of some of the most important generalized inverses. We also present generalizations of the other results of these authors for the case of the Moore-Penrose inverse. © 2011 Published by Elsevier Inc.

Nonnegative generalized inverses and least elements of polyhedral sets

We propose new notions of monotonicity for real matrices in order to characterize a particular notion of interval boundedness of various generalized inverses. © 2010 Warsaw University. All rights reserved.

Demonstratio Mathematica

Matrix interval monotonicity

In this article, a new characterization for the nonnegativity of Moore-Penrose inverses of Gram operators over Hilbert spaces is presented. The main result generalizes the existing result for invertible Gram operators. © 2008 Birkhäuser Verlag Basel/Switzerland.

Positivity

A new characterization of nonnegativity of Moore-Penrose inverses of Gram operators

Infinite matrices, the forerunner and a main constituent of many branches of classical mathematics (infinite quadratic forms, integral equations, differential equations, etc.) and of the modern operator theory, is revisited to demonstrate its deep influence on the development of many branches of mathematics, classical and modern, replete with applications. This review does not claim to be exhaustive, but attempts to present research by the authors in a variety of applications. These include the theory of infinite and related finite matrices, such as sections or truncations and their relationship to the linear operator theory on separable and sequence spaces. Matrices considered here have special structures like diagonal dominance, tridiagonal, sign distributions, etc. and are frequently nonsingular. Moreover, diagonally dominant finite and infinite matrices occur largely in numerical solutions of elliptic partial differential equations.The main focus is the theoretical and computational aspects concerning infinite linear algebraic and differential systems, using techniques like conformal mapping, iterations, truncations etc. to derive estimates based solutions. Particular attention is paid to computable precise error estimates, and explicit lower and upper bounds. Topics include Bessel's, Mathieu equations, viscous fluid flow, simply and doubly connected regions, digital dynamics, eigenvalues of the Laplacian, etc. Also presented are results in generalized inverses and semi-infinite linear programming. © 2008 Elsevier Inc. All rights reserved.

A review of infinite matrices and their applications

Let X be a partially ordered real Banach space, let a,b X with ab. Let φ be a bounded linear functional on X. We say that X satisfies the box-optimization property (or X is a BOP space) if the box-constrained linear program: max aφ,x, s.t. aaxb, has an optimal solution for any φ,a and b. Such problems arise naturally in solving a class of problems known as interval linear programs. BOP spaces were introduced (in a different language) and systematically studied in the first author's doctoral thesis. In this paper, we identify new classes of Banach spaces that are BOP spaces. We present also sufficient conditions under which answers are in the affirmative for the following questions: (i) When is a closed subspace of a BOP space a BOP space? (ii) When is the range of a bounded linear map a BOP space? (iii) Is the quotient space of a BOP space a BOP space? © 2008 Springer Science+Business Media, LLC.

Journal of Optimization Theory and Applications

Linear optimization with box constraints in banach spaces

In this paper, we derive necessary and sufficient conditions for the nonnegativity of the Moore-Penrose inverse of a Gram matrix defined in an indefinite inner product space using indefinite matrix multiplication. These conditions include the acuteness of a pair of closed convex cones. © 2008 Springer Science+Business Media, LLC.

Nonnegative Moore-Penrose inverse of Gram matrices in an indefinite inner product space

Theorems of the alternative over indefinite inner product spaces

Let M=[A a] be a matrix of order m×n, where A ∈ ℝ m×(n-1) and a ∈ ℝm is an m×1 vector. In this article, we derive a formula for the Moore-Penrose inverse of M * M and obtain sufficient conditions for its nonnegativity. The results presented here generalize the ones known earlier. © 2008 Springer Science+Business Media, LLC.

Moore-Penrose inverse of a gram matrix and its nonnegativity

The objective of this paper is to study the nonnegativity of the Moore-Penrose inverse of an operator between real Hilbert spaces. A sufficient condition ensuring this is given in terms of certain spectral property of all positive splittings of the given operator. A partial converse is proved. © 2008 Springer Science + Business Media B.V.

Nonnegative moore-penrose inverses of operators over Hilbert spaces

This short note presents a sufficient condition under which a question raised by Peris and Subiza, namely, to find a {1}-inverse X ∈ Rn × m of a matrix A ∈ Rm × n such that A is weak monotone if and only if X is nonnegative on R (A), is answered in the affirmative. © 2007 Elsevier Inc. All rights reserved.

Weak monotonicity of A versus {1}-inverses nonnegative on the range space of A

This paper is concerned with necessary and sufficient conditions for the nonnegativity of Moore-Penrose inverses of Gram operators between real Hilbert spaces. These conditions include statements on acuteness (or obtuseness) of certain closed convex cones. The main result generalizes a well known result for inverses in the finite dimensional case over the nonnegative orthant to Moore-Penrose inverses in (possibly) infinite dimensional Hilbert spaces over any general closed convex cone. © 2006 Elsevier Inc. All rights reserved.

Nonnegative Moore-Penrose inverses of Gram operators

Let H be a Hilbert space with {en : n ∈ N} as an orthonormal basis. Let T : H → H be a bounded linear operator defined by Ten = en - 1 + λ sin (2 nr) en + en + 1, where λ is real and r is a rational multiple of π. In this short note it is established that the Moore-Penrose inverse of T is not bounded. We also show that the same conclusion is valid for a few related classes of operators. © 2006 Elsevier Inc. All rights reserved.

Applied Mathematics and Computation

Generalized inverses of tridiagonal operators

Let H1 and H2 be indefinite inner product spaces. Let L (H1) and L (H2) be the sets of all linear operators on H1 and H2, respectively. The following result is proved: If Φ is [*]-isomorphism from L (H1) onto L (H2) then there exists U : H1 → H2 such that Φ (T) = c U T U[*] for all T ∈ L (H1) with U U[*] = c I2, U[*] U = c I1 and c = ± 1. Here I1 and I2 denote the identity maps on H1 and H2, respectively. © 2006 Elsevier Inc. All rights reserved.

Journal of Mathematical Analysis and Applications

A characterization for *-isomorphisms in an indefinite inner product space

Three-level inversion can be achieved with dual two-level inverters feeding an open-end winding induction motor. A space vector-based pulse width-modulated (PWM) switching scheme is proposed, which achieves three-level inversion using all the space vector locations with a single DC power supply. The zero-sequence current is reduced by reducing the zero-sequence voltage to zero, in the average sense, within a sampling time interval. The proposed PWM scheme balances the zero-sequence current dynamically, simply with a relocation of the effective time within a sampling time interval. The switching frequency of the inverters is also reduced to half as only one inverter is switched while the other is clamped in a sampling time interval. Also, the proposed PWM scheme employs only the three instantaneous phase reference voltages obviating the necessity of sector identification and lookup tables.

Department of Electrical Engineering

IET Digital Library

Pulse width-modulated switching strategy for the dynamic balancing of zero-sequence current for a dual-inverter fed open-end winding induction motor drive

Recently, it was shown that, contrary to the case of finite matrices (with real or complex entries) an invertible infinite matrix V could have a Moore-Penrose inverse that is not a classical inverse of V. In this article, we show that V has infinitely many group inverses that are also Moore-Penrose inverses.

Generalized inverses of an invertible infinite matrix

In this article we show that, contrary to finite matrices (with real or complex entries) an invertible infinite matrix V could have a Moore-Penrose inverse that is not a classical inverse of V . This also answers a recent open problem on infinite matrices.