Research Letters in the Information and Mathematical Sciences

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Research Letters welcomes papers from staff and graduate students at Massey University in the areas of: Computer Science, Information Science, Mathematics, Statistics and the Physical and Engineering Sciences. Research letters is a preprint series that accepts articles of completed research work, technical reports, or preliminary results from ongoing research. After editing, articles are published online and can be referenced, or handed out at conferences. Copyright remains with the authors and the articles can be used as preprints to academic journal publications or handed out at conferences. Editors Dr Elena Calude Dr Napoleon Reyes The guidelines for writing a manuscript can be accessed here.

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Author: search.filters.author.Rao, RadhakrishnaSubject: search.filters.subject.Eigenvector

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    Antieigenvalues and antisingularvalues of a matrix and applications to problems in statistics
    (Massey University, 2005) Rao, Radhakrishna
    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.