## 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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