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

Simple item page
dc.contributor.authorRao, Radhakrishna
dc.date.accessioned2013-05-15T03:37:27Z
dc.date.available2013-05-15T03:37:27Z
dc.date.issued2005
dc.description.abstractLet 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.citationRao, R. (2005), Antieigenvalues and antisingularvalues of a matrix and applications to problems in statistics, Research Letters in the Information and Mathematical Sciences, 8, 53-76en
dc.identifier.issn1175-2777
dc.identifier.urihttp://hdl.handle.net/10179/4452
dc.language.isoenen
dc.publisherMassey Universityen
dc.subjectEigenvectoren
dc.subjectEigenvalueen
dc.subjectAnti-eigenvectoren
dc.subjectAnti-eigenvalueen
dc.subjectMatrix (mathematics)en
dc.titleAntieigenvalues and antisingularvalues of a matrix and applications to problems in statisticsen
dc.typeArticleen
Files
Original bundle
Now showing 1 - 1 of 1
Thumbnail Image
Name:
Antieigenvalues_and_antisingularvalues_of_a_matrix_and_applications_to_problems_in_statistics.pdf
Size:
187.46 KB
Format:
Adobe Portable Document Format
Description:
Download
License bundle
Now showing 1 - 1 of 1
Thumbnail Image
Name:
license.txt
Size:
1.71 KB
Format:
Item-specific license agreed upon to submission
Description:
Download
Collections
Research Letters in the Information and Mathematical Sciences