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Assessing the Effects of Multiple Rows on the Condition Number of a Matrix

Ali S. Hadi and Martin T. Wells
Journal of the American Statistical Association
Vol. 85, No. 411 (Sep., 1990), pp. 786-792
DOI: 10.2307/2290016
Stable URL: http://www.jstor.org/stable/2290016
Page Count: 7
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Assessing the Effects of Multiple Rows on the Condition Number of a Matrix
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Abstract

Given a matrix X of n observations on k variables, it is known that the singular values of X can change substantially when few rows are omitted from X. Hadi (1988) shows that no general closed-form equation can relate the singular values of X to the singular values of X with one row deleted and gives closed-form approximations to the relationship between the singular values of the two matrices. In this article, we extend Hadi's results to the more general case of multiple-row deletion, carry out systematic numerical investigations to determine the goodness and the speed of the approximation, and give an example using real-life data to illustrate the usefulness of the results in diagnosing jointly influential observations in linear regression. The methods presented in this article deal with the computational as well as the data analytic aspects of problems arising in multivariate data analysis. These methods are applicable in situations where the eigenstructure of a matrix is of interest.

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