Now showing 1 - 10 of 76
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    M-operators on partially ordered Banach spaces
    (01-01-2019)
    Kalauch, A.
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    Lavanya, S.
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    For a matrix A ∈ ℝn × n whose off-diagonal entries are nonpositive, there are several well-known properties that are equivalent to A being an invertible M-matrix. One of them is the positive stability of A. A generalization of this characterization to partially ordered Banach spaces is considered in this article. Relationships with certain other equivalent conditions are derived. An important result on singular irreducible M-matrices is generalized using the concept of M-operators and irreducibility. Certain other invertibility conditions of M-operators are also investigated.
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    Vanishing pseudo–Schur complements, reverse order laws, absorption laws and inheritance properties
    (02-01-2018)
    Bisht, Kavita
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    The problem of vanishing of a (generalized) Schur complement of a block matrix (corresponding to the leading principal subblock) implying that the other (generalized) Schur complement (corresponding to the trailing principal subblock) is zero, is revisited. Absorption laws for two important classes of generalized inverses are considered next. Inheritance properties of the generalized Schur complements in relation to the absorption laws are derived. Inheritance by the generalized principal pivot transform is also studied.
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    On the matrix class Q0 and inverse monotonicity properties of bordered matrices
    (01-03-2021)
    Choudhury, Projesh Nath
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    Eagambaram, N.
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    Sushmitha, P.
    Let A be a square matrix satisfying either a monotonicity condition (like inverse nonnegativity) or a property expressed in terms of the existence of solution of the linear complementarity problem, defined using A (like the Q0-property). We address the question of when the bordered matrix M, whose leading (maximal) proper principal submatrix being the matrix A, inherits the said property of A.
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    Almost definite matrices revisited
    (01-01-2015)
    Meenakshi, Ar R.
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    Choudhury, Projesh Nath
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    A real matrix A is called as an almost definite matrix if 〈x,Ax〉 = 0 ⇒ Ax = 0. This notion is revisited. Many basic properties of such matrices are established. Several characterizations for a matrix to be an almost definite matrix are presented. Comparisons of certain properties of almost definite matrices with similar properties for positive definite or positive semidefinite matrices are brought to the fore. Interconnections with matrix classes arising in the theory of linear complementarity problems are discussed briefly.
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    A class of singular R0-matrices and extensions to semidefinite linear complementarity problems
    (29-08-2013)
    For A ∈ ℝn×n and q ∈ ℝn, the linear complementarity problem LCP(A; q) is to determine if there is x ∈ ℝn such that x ≥ 0; y = Ax+q ≥ 0 and xT y = 0. Such an x is called a solution of LCP(A; q). A is called an R 0-matrix if LCP(A; 0) has zero as the only solution. In this article, the class of R0-matrices is extended to include typically singular matrices, by requiring in addition that the solution x above belongs to a subspace of ℝn. This idea is then extended to semidefinite linear complementarity problems, where a characterization is pre-sented for the multplicative transformation.
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    Some results on a class of optimization spaces
    (01-01-2005) ;
    Swarna, J. Mercy
    Let X be a partially ordered real Banach space, a, b ∈ X with a ≤ b. Let φ be a bounded linear functional on X. We call X a Ben-Israel-Charnes space (or a B-C space, for short) if the linear program: Maximize 〈φ, x〉 subject to a ≤ x ≤ b has an optimal solution. Such problems have been shown to be important in solving a class of problems known as Interval Linear Programs. B-C spaces were introduced by the first author in his doctoral dissertation. In this paper we identify new classes of Banach spaces that are B-C spaces. We also present sufficient conditions under which answers are in the affirmative for the following questions: 1 When is a closed subspace of a B-C space, a, B-C space? 2 Is the range of a bounded linear map from a Banach space into a B-C space, a B-C space? © Springer-Verlag Berlin Heidelberg 2005.
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    Singular irreducible M-operators on ordered Banach spaces
    (01-06-2021)
    Kalauch, A.
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    Lavanya, S.
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    This work continues our earlier work on singular M-operators for operators over ordered Banach spaces, where we show how some interesting results on singular irreducible M-matrices have analogues to operators over ordered Banach spaces.
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    A dominance notion for singular matrices with applications to nonnegative generalized inverses
    (01-08-2012)
    Mishra, Debasisha
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    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.
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    On nonnegative moore-penrose inverses of perturbed matrices
    (11-06-2013)
    Jose, Shani
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    Nonnegativity of the Moore-Penrose inverse of a perturbation of the form A-XGYT is considered when A†≥0. Using a generalized version of the Sherman-Morrison-Woodbury formula, conditions for (A-XGYT) † to be nonnegative are derived. Applications of the results are presented briefly. Iterative versions of the results are also studied. © 2013 Shani Jose and K. C. Sivakumar.