Roger A. Horn

Matrix analysis: Frontmatter

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.

The Relationship Between Severity of Illness and Hospital Length of Stay and Mortality

To address the question of quantification of severity of illness on a wide scale, the Computerized Severity Index (CSI) was developed by a research team at the Johns Hopkins University. This article describes an initial assessment of some aspects of the validity and reliability of the CSI on a sample of 2,378 patients within 27 high-volume DRGs from five teaching hospitals. The 27 DRGs predicted 27% of the variation in LOS, while DRGs adjusted for Admission CSI scores predicted 38% and DRGs adjusted for Maximum CSI scores throughout the hospital stay predicted 54% of this variation. Thus, the Maximum CSI score increased the predictability of DRGs by 100%. We explored the impact of including a 7-day cutoff criterion along with the Maximum CSI score similar to a criterion used in an alternative severity of illness measure. The DRG/Maximum CSI scores predictive power increased to 63% when the 7-day cutoff was added to the CSI definition. The Admission CSI score was used to predict in-hospital mortality and correlated R = 0.603 with mortality. The reliability of Admission and Maximum CSI data collection was high, with agreement of 95% and kappa statistics of 0.88 and 0.90, respectively.

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

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 Administrations 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.

Estimating Heteroscedastic Variances in Linear Models

Abstract We describe an estimator of heteroscedastic variances in the Gauss-Markov linear model where E(e) = 0 and with σ i 2 and unknown. It may be thought of as an approximation to the MINQUE method which results in computational economy, positive estimates, and decreased mean square error. Properties of this almost unbiased estimator are stated. It is compared with other estimators, and extensions to more general models are discussed.

Canonical forms for complex matrix congruence and ∗congruence

Abstract Canonical forms for congruence and ∗congruence of square complex matrices were given by Horn and Sergeichuk in [Linear Algebra Appl. 389 (2004) 347–353], based on Sergeichuk’s paper [Math. USSR, Izvestiya 31 (3) (1988) 481–501], which employed the theory of representations of quivers with involution. We use standard methods of matrix analysis to prove directly that these forms are canonical. Our proof provides explicit algorithms to compute all the blocks and parameters in the canonical forms. We use these forms to derive canonical pairs for simultaneous congruence of pairs of complex symmetric and skew-symmetric matrices as well as canonical forms for simultaneous ∗congruence of pairs of complex Hermitian matrices.

A canonical form for matrices under consimilarity

Abstract Square complex matrices A , B are said to be consimilar if A=SB S −1 for some nonsingular matrix S . Consimilarity is an equivalence relation that is a natural matrix generalization of rotation of scalars in the complex plane. We survey the known forms to which a given complex matrix may be reduced by unitary consimilarity and describe a canonical form to which it may be reduced by a general consimilarity. We derive a useful criterion for two matrices to be consimilar and show that every matrix is consimilar to its own conjugate, transpose, and adjoint, to a real matrix, and to a Hermitian matrix.

The computerized severity index: a new tool for case-mix management

We describe the new Computerized Severity Index (CSI) that is obtained from an expanded discharge abstract data set, based on a 6th-digit severity addition to the ICD-9-CM coding system. The new 6-digit code book (called ICD-9-CMSA) is used to label existence and severity of each principal and secondary diagnosis. It can be used to produce an overall severity of illness level for each hospital inpatient. The impact of severity-adjusted DRGs on prospective payment and uses of the CSI for assessing quality of care, efficiency, physician practice profiles, and prediction of posthospital resource needs are discussed.

Congruences of a square matrix and its transpose

Abstract It is known that any square matrix A over any field is congruent to its transpose: AT=STAS for some nonsingular S; moreover, S can be chosen such that S2=I, that is, S can be chosen to be involutory. We show that A and AT are ∗ congruent over any field F of characteristic not two with involution a↦ a (the involution can be the identity): A T = S T AS for some nonsingular S; moreover, S can be chosen such that S S=I , that is, S can be chosen to be coninvolutory. The short and simple proof is based on Sergeichuks canonical form for ∗ congruence [Math. USSR, Izvestiya 31 (3) (1988) 481]. It follows that any matrix A over F can be represented as A=EB, in which E is coninvolutory and B is symmetric.

Psychiatric severity of illness. A case mix study.

This study was undertaken to determine if a measure of severity of illness for psychiatric patients, the Psychiatric Severity of Illness Index, could produce psychiatric case mix groups that are more homogeneous with respect to resource use than the diagnosis-related groups (DRGs). Psychiatric Severity of Illness data were collected on 1,672 cases in ten hospitals of various types. Of these cases, 1,418 had enough information in the medical record to be scored using the Psychiatric Severity Index, 1,173 of which were in MDC 19 (mental diseases and disorders). We found that four Psychiatric Severity of Illness groups explained between 34% and 50% of the variation in length of stay of the combined hospital data in MDC 19, whereas nine DRGs explained between 6% and 14%. DRGs subdivided by Psychiatric Severity of Illness groups explained between 40% and 54% of the variation in length of stay. The implications of these results for cross-hospital comparisons are discussed.

Severity of illness within DRGs. Homogeneity study.

The authors assess the ability of the Severity of Illness Index to explain variability of resource use within each DRG. The data came from 15 hospitals, all of which had a HCFA DRG case mix index greater than 1. The data set comprised approximately 106,000 discharges, for which discharge abstract data, financial data, and Severity of Illness data were available. To pool the data over the 15 hospitals, the authors converted all charges to costs and normalized them to fiscal year 1983. Adjustments were also made for medical education and wage levels. The Severity of Illness Index explained more than 10% of the variability in resource use in 94% of the DRGs, which contained 97% of the patients, and more than 50% of the variability in resource use in 36% of the DRGs, which contained 24% of the patients. For the whole data set, DRGs explained 28% of the variability in resource use, and severity-adjusted DRGs explained 61% of the variability in resource use. Thus the Severity of Illness Index explained a large amount of the variability in resource use within individual DRGs as well as in the whole data set. This explanatory power remained when outliers were removed. These results go beyond previous studies that were based on six disease conditions and/or were analyzed only within individual hospitals. The findings indicate that the phenomenon of severity of illness differences within DRGs, and the corresponding differences in resource use, is consistent across 15 hospitals that represent all sections of the United States and all teaching types.