Showing papers by "Roger A. Horn published in 1985"

Matrix Analysis

Abstract: Linear algebra and matrix theory are fundamental tools in mathematical and physical science, as well as fertile fields for research. This new edition of the acclaimed text presents results of both classic and recent matrix analyses using canonical forms as a unifying theme, and demonstrates their importance in a variety of applications. The authors have thoroughly revised, updated, and expanded on the first edition. The book opens with an extended summary of useful concepts and facts and includes numerous new topics and features, such as: - New sections on the singular value and CS decompositions - New applications of the Jordan canonical form - A new section on the Weyr canonical form - Expanded treatments of inverse problems and of block matrices - A central role for the Von Neumann trace theorem - A new appendix with a modern list of canonical forms for a pair of Hermitian matrices and for a symmetric-skew symmetric pair - Expanded index with more than 3,500 entries for easy reference - More than 1,100 problems and exercises, many with hints, to reinforce understanding and develop auxiliary themes such as finite-dimensional quantum systems, the compound and adjugate matrices, and the Loewner ellipsoid - A new appendix provides a collection of problem-solving hints.

Interhospital differences in severity of illness. Problems for prospective payment based on diagnosis-related groups (DRGs).

Abstract: We evaluated the ability of the diagnosis-related-group (DRG) classification system to account adequately for severity of illness and, by implication, for the costs of medical care. Hospital inpatients on medicine, surgery, obstetrics/gynecology, and pediatrics services in six hospitals were evaluated to provide a spectrum of patient and hospital characteristics. This evaluation was based on data from a generic index of severity of illness obtained by trained personnel from a review of hospital charts after patient discharge. Within each DRG, substantial differences were found in the distribution of severity of illness in different hospitals. Some hospitals treated larger proportions of severely ill patients and had a wide range of severity within each DRG, but these differences did not always agree with the teaching classification or the Health Care Financing Administration's case-mix index. These findings suggest that patient classification by means of unadjusted DRGs does not adequately reflect severity of illness, and they indicate that prospective payment programs based on DRGs alone may unfairly and adversely discriminate against certain hospitals.

Severity of illness within DRGs: impact on prospective payment.

Abstract: This study compares the financial impact of a Diagnosis Related Group (DRG) prospective payment system with that of a Severity of Illness-adjusted DRG prospective payment system. The data base of about 106,000 discharges is from 15 hospitals, all of which had a Health Care Financing Administration (HCFA) DRG case mix index greater than 1. In order to pool the data over the 15 hospitals, all charges were converted to costs, normalized to Fiscal Year 1983, and adjusted for medical education and wage levels. The findings showed that, for the study population as a whole, DRGs explained 28 per cent of the variability in resource use per case while Severity of Illness-adjusted DRGs explained 61 per cent of the variability in resource use per case. When we simulated prospective payment systems based on DRGs and on Severity-adjusted DRGs, we found that the financial impact of the two systems differed by very little in some hospitals and by as much as 35 per cent of total operating costs in other hospitals. Thus, even with a data set that is relatively homogeneous (with respect to the HCFA DRG case mix index definition of hospitals), we found substantial inequities in payment when DRGs were not adjusted for Severity of Illness. These findings suggest that, with a more representative set of hospitals, the difference between unadjusted and Severity-adjusted DRG-based prospective payment could be greater than 35 per cent of a hospital's total operating costs.

Norms for vectors and matrices

Abstract: In linear algebra, we work with vectors and matrices. Vectors are our primary objects, and we think of a matrix as an operator that can transform one vector into another. Vectors and matrices are lists and tables of numbers, so it may not be obvious how to say that one object is large and another small. We will define norms that measure every object, allowing us to make meaningful statements about size, comparisons, and convergence.

On simultaneous reduction of families of matrices to triangular or diagonal form by unitary congruences

Abstract: Let {Ai }and {Bi } be two given families of n-by-n matrices. We give conditions under which there is a unitary U such that every matrix UAiU 1 is upper triangular. We give conditions, weaker than the classical conditions of commutativity of the whole family, under which there is a unitary U such that every matrix UAjU ∗ is upper triangular. We also give conditions under which there is one single unitary U such that every UAiU 1 and every UBjU ∗ is upper triangular. We give necessary and sufficient conditions for simultaneous unitary reduction to diagonal form in this way when all the Aj's are complex symmetric and all theBj 's are Hermitian.

Matrix analysis: Location and perturbation of eigenvalues

Abstract: Introduction The eigenvalues of a diagonal matrix are very easy to locate, and the eigenvalues of a matrix are continuous functions of the entries, so it is natural to ask whether one can say anything useful about the eigenvalues of a matrix whose off-diagonal elements are “small” relative to the main diagonal entries. Such matrices do arise in practice; large systems of linear equations resulting from numerical discretization of boundary value problems for elliptic partial differential equations can be of this form. In some differential equations problems involving the long-term stability of an oscillating system, one is sometimes interested in showing that the eigenvalues {λ i } of a matrix all lie in the left half-plane, that is, that Re(λ i ) i > 0. Sometimes one wants to locate the eigenvalues of a matrix in a bounded set that is easily characterized. We know that all the eigenvalues of a matrix A are located in a disc in the complex plane centered at the origin and having radius ∥ A ∥, where ∥·∥ is any matrix norm. But can one do better than this by more precisely locating regions that must either include or exclude the eigenvalues? We shall see that one can.