On group inverses and the sharp order

AbstractIn 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

On parallel sum and difference of matrices

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Generalized inverse of a matrix and its applications

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Conditions for optimality and validity of simple least squares theory

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Reducing the environmental impact of surgery on a global scale: systematic review and co-prioritization with healthcare workers in 132 countries

Abstract Background Healthcare cannot achieve net-zero carbon without addressing operating theatres. The aim of this study was to prioritize feasible interventions to reduce the environmental impact of operating theatres. Methods This study adopted a four-phase Delphi consensus co-prioritization methodology. In phase 1, a systematic review of published interventions and global consultation of perioperative healthcare professionals were used to longlist interventions. In phase 2, iterative thematic analysis consolidated comparable interventions into a shortlist. In phase 3, the shortlist was co-prioritized based on patient and clinician views on acceptability, feasibility, and safety. In phase 4, ranked lists of interventions were presented by their relevance to high-income countries and low–middle-income countries. Results In phase 1, 43 interventions were identified, which had low uptake in practice according to 3042 professionals globally. In phase 2, a shortlist of 15 intervention domains was generated. In phase 3, interventions were deemed acceptable for more than 90 per cent of patients except for reducing general anaesthesia (84 per cent) and re-sterilization of ‘single-use’ consumables (86 per cent). In phase 4, the top three shortlisted interventions for high-income countries were: introducing recycling; reducing use of anaesthetic gases; and appropriate clinical waste processing. In phase 4, the top three shortlisted interventions for low–middle-income countries were: introducing reusable surgical devices; reducing use of consumables; and reducing the use of general anaesthesia. Conclusion This is a step toward environmentally sustainable operating environments with actionable interventions applicable to both high– and low–middle–income countries

A new class of g-inverse of square matrices

Necessary and sufficient conditions are obtained for a matrix A to have a g-inverse with rows and columns belonging to special linear manifolds. For a square matrix A, a g-inverse, with columns belonging to the linear manifold generated by the columns of A, is denoted by A-C. Such a g-inverse exists if and only if R(A)=R(A2). The following properties of A-C are established: (a) A-C= A(A2)-. (b) For any positive integer m, (A-C)m provides a reflexive g-inverse of Am. (c) If x is an eigenvector corresponding to a nonnull eigenvalue λ of A, x is also an eigenvector of A-C corresponding to its eigenvalue 1/λ. The converse of this result is also true. (d) A special choice of (A2)-=(A3)-A leads to A-C=A(A3)-A which is unique irrespective of the choice of (A3)- and is, in fact, the same as the Scroggs-Odell pseudoinverse (J.SIAM 1966) of A. When R(A)=R(A2), this indeed is a much simpler way of calculating the Scroggs-Odell pseudoinverse compared to the method indicated by its authors. (e) A(A3)-A belongs to the subalgebra generated by A

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

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

Some remarks on the missing plot analysis

The analysis of variance of incomplete data from randomized block and Latin square experiments is considered and the expected values, under the null hypothesis, of the treatment and error mean squares are obtained. For simplicity, only the case of a single missing observation is considered. The results could be similarly extended to the case of multiple missing observations in a more general situation where the null hypothesis need not be true