## Antieigenvalues and antisingularvalues of a matrix and applications to problems in statistics

 dc.contributor.author Rao, Radhakrishna dc.date.accessioned 2013-05-15T03:37:27Z dc.date.available 2013-05-15T03:37:27Z dc.date.issued 2005 dc.description.abstract Let A be p × p positive definite matrix. A p-vector x such that Ax = x is called an eigenvector with the associated with eigenvalue . Equivalent characterizations are: (i) cos = 1, where is the angle between x and Ax. (ii) (x0Ax)−1 = xA−1x. (iii) cos = 1, where is the angle between A1/2x and A−1/2x. We ask the question what is x such that cos as defined in (i) is a minimum or the angle of separation between x and Ax is a maximum. Such a vector is called an anti-eigenvector and cos an anti-eigenvalue of A. This is the basis of operator trigonometry developed by K. Gustafson and P.D.K.M. Rao (1997), Numerical Range: The Field of Values of Linear Operators and Matrices, Springer. We may define a measure of departure from condition (ii) as min[(x0Ax)(x0A−1x)]−1 which gives the same anti-eigenvalue. The same result holds if the maximum of the angle between A1/2x and A−1/2x as in condition (iii) is sought. We define a hierarchical series of anti-eigenvalues, and also consider optimization problems associated with measures of separation between an r(< p) dimensional subspace S and its transform AS. Similar problems are considered for a general matrix A and its singular values leading to anti-singular values. Other possible definitions of anti-eigen and anti-singular values, and applications to problems in statistics will be presented. en dc.identifier.citation Rao, R. (2005), Antieigenvalues and antisingularvalues of a matrix and applications to problems in statistics, Research Letters in the Information and Mathematical Sciences, 8, 53-76 en dc.identifier.issn 1175-2777 dc.identifier.uri http://hdl.handle.net/10179/4452 dc.language.iso en en dc.publisher Massey University en dc.subject Eigenvector en dc.subject Eigenvalue en dc.subject Anti-eigenvector en dc.subject Anti-eigenvalue en dc.subject Matrix (mathematics) en dc.title Antieigenvalues and antisingularvalues of a matrix and applications to problems in statistics en dc.type Article en
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