K.C. Sivakumar

Comparison theorems for a subclass of proper splittings of matrices

Abstract In this article, a convergence theorem and several comparison theorems are presented for a subclass of proper splittings of matrices introduced recently.

Applications of generalized inverses to interval linear programs in hilbert spaces

Let H1 and H2 be real Hilbert spaces. Suppose H2 is partially ordered, a, b ∊ H2 c ∊ H1 and A :H1 → H2 is a continuous linear map. We consider the following interval linear program: Maximize subject to a ≤ Ax ≤ b. Conditions under which explicit solutions to the above problem can be found are studied. The solutions are represented in terms of generalized inverses of A. Several examples are given to illustrate the main results.

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

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.

On partial orders of Hilbert space operators

First, an overview of partial orders defined on bounded linear operators on an infinite-dimensional Hilbert space is presented. A definition for the core inverse of operators on a Hilbert space is proposed. Extensions of the sharp and the core partial orders are considered. An explicit formula for the core inverse of matrices is obtained using a full-rank factorization. Relationships between all these partial orders and formula for generalized inverses of differences of operators, when they are related with respect to these partial orders, are investigated.

P†-matrices: a generalization of P-matrices

For , let r(A,B) (c(A,B)) be the set of matrices whose rows, (columns) are independent convex combinations of the rows (columns) of . Johnson and Tsatsomeros have shown that the set r(A,B) (c(A,B)) consists entirely of nonsingular matrices if and only if is a -matrix. For , let . Rohn has shown that if all the matrices in are invertible, then , and are -matrices. In this article, we define a new class of matrices called -matrices and present certain extensions of the above results to the singular case, where the usual inverse is replaced by the Moore–Penrose generalized inverse. The case of the group inverse is briefly discussed.

On Certain Positivity Classes of Operators

ABSTRACT A real square matrix A is called a P-matrix if all its principal minors are positive. Such a matrix can be characterized by the sign non-reversal property. Taking a cue from this, the notion of a P-operator is extended to infinite dimensional spaces as the first objective. Relationships between invertibility of some subsets of intervals of operators and certain P-operators are then established. These generalize the corresponding results in the matrix case. The inheritance of the property of a P-operator by the Schur complement and the principal pivot transform is also proved. If A is an invertible M-matrix, then there is a positive vector whose image under A is also positive. As the second goal, this and another result on intervals of M-matrices are generalized to operators over Banach spaces. Towards the third objective, the concept of a Q-operator is proposed, generalizing the well known Q-matrix property. An important result, which establishes connections between Q-operators and invertible M-operators, is proved for Hilbert space operators.

Extensions of Perron–Frobenius splittings and relationships with nonnegative Moore–Penrose inverses

An -matrix has the form , where and is eventually nonnegative, i.e. is entry wise nonnegative for all sufficiently large integers . In this article, two new types of splittings of matrices are introduced. The class of matrices possessing a splitting of one of these types includes -matrices as a subclass. The authors derived necessary and sufficient conditions for the convergence of these splittings in terms of an extended notion of the nonnegativity of the Moore–Penrose inverse. The work reported here widens the applicability of the existing results.

Matrix interval monotonicity

We propose new notions of monotonicity for real matrices in order to characterize a particular notion of interval boundedness of various generalized inverses. AMS Subject Classification: 15A09, 15A48.

Vanishing pseudo–Schur complements, reverse order laws, absorption laws and inheritance properties

Abstract 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.

Inheritance and inverse monotonicity properties of copositive matrices

A symmetric matrix is called copositive if it satisfies the inequality whenever and strictly copositive if , whenever . The ordering of a vector here is component-wise. Certain interesting properties of the inverse of a copositive matrix are extended to its Moore–Penrose inverse. The inheritance property of the Schur complement of a copositive matrix is extended to the case when the inverses in the Schur complement are replaced by their Moore–Penrose inverses. A framework is provided wherein one has the copositivity of , given the copositivity of .