Sujit Kumar Mitra

A pair of simultaneous linear matrix equations A1XB1 = C1, A2XB2 = C2 and a matrix programming problem

Necessary and sufficient conditions are obtained for a pair of matrix equations A1XB1 = C1, A2XB2 = C2 on a general field F to have a common solution, along with the expression for a general common solution when certain conditions hold. Common solutions of minimum rank are described. A matrix programming problem is solved en route.

The matrix equations AX = C, XB = D

For the pair of matrix equations AX = C, XB = D this paper gives common solutions of minimum possible rank and also other feasible specified ranks.

Matrix Partial Orders, Shorted Operators and Applications

Introduction Decompositions and Generalized Inverses Minus Order Sharp Order Star Order One-Sided Orders Lowner Order and Majorization Unified Theory of Matrix Partial Orders through Generalized Inverses Parallel Sums Schur Complements and Shorted Operators Shorted Operators II Supremum and Infimum for a Pair of Matrices Partial Orders for Modified Matrices Statistics Electrical Network Theory.

Common solutions to a pair of linear matrix equations A 1 XB 1 = C 1 and A 2 XB 2 = C 2

Penrose (4) gave a necessary and sufficient condition for the consistency of the linear matrix equation AXB = C and also its complete class of solutions. A necessary and sufficient condition for the equations AX = C, XB = D to have a common solution was given by Cecioni (3) and an expression for the general common solution by Rao and Mitra ((6), p. 25). In the present paper, we obtain a necessary and sufficient condition for the equations A1XB1 = C1 and A2XB2 = C2 to have a common solution and also an expression for the general common solution. This result isuseful in computing a constrained inverse of a matrix, a concept originallyintroduced by Bott and Duffin(2) and recently extended by Rao and Mitra(7) who consider more general constraints with the object of bringing together the various generalized inverses and pseudoinverses under a common classification scheme.

On group inverses and the sharp order

Abstract In this work the group inverse of a matrix is used to define the #-order on square matrices of index 1. The #-order is similar to the ∗-order of Drazin [2] and the minus order of Hartwig [6, 10] and Nambooripad [17]. The #-order and the ∗-order are compared and contrasted. Many conditions are given which assure the equivalence of the various partial orders studied.

Left-star and right-star partial orderings

Abstract Two partial orderings in the set of complex matrices are introduced by combining each of the conditions A*A = A*B and AA* = BA*, which define the star partial ordering, with one of the conditions M (A) ⊆ M (B) and M (A*) ⊆ M (B*), which define the space preordering. Several properties of these orderings are examined, with main emphasis on comparing the new orderings with the star ordering, the minus ordering, and other related partial orderings. Moreover, some further characterizations of partial orderings in terms of inclusions of appropriate classes of generalized inverses are derived, with the main emphasis on characterizations involving reflexive generalized inverses.

Projections under seminorms and generalized Moore Penrose inverses

The definition of a projector under a seminorm is given. Such a projector is not unique. Operators projecting into a given linear subspace under a seminorm form an affine linear subalgebra of the linear associative algebra of square matrices. The authors have introduced elsewhere the concept of a minimum seminorm semileast squares inverse of a complex matrix. It is shown here that the same concept could also be defined in terms of projectors under seminorms. This extends a similar definition for the Moore Penrose inverse given in terms of orthogonal projectors under the usual Euclidean norms. Various properties of a projector under a seminorm and also of a minimum seminorm semileast squares inverse are obtained including representations giving general solutions for both.

The minus partial order and the shorted matrix

The minus partial order was defined by Hartwig [6], weakening the conditions of the star partial order of Drazin [5]. Several new properties of the minus partial order are established. The minus partial order is used to redefine the shorted matrix [15, 17] and to define the infimum A ∧ B and the supremum A ∨ B of a pair A, B of matrices of the same order. The definition of the shorted matrix given here is similar to the Krein-Anderson-Trapp definition of the shorted positive operator.

Partial Orders Based on Outer Inverses

Classes of outer inverses are used to create new partial orders and unify the classification of some of the old partial orders. These classes are then compared with those obtained by using g-inverses (inner inverses).

Matrix partial orders through generalized inverses: unified theory

The unified theory presented here covers as special cases the star order of Drazin, the minus order of Hartwig and Nambooripad, the sharp order, and other partial orders introduced by the author.