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

Abstract

Let A be p×p positive definite matrix. A p-vector x such that Ax=λx is called an eigenvector with the associated with eigenvalue λ. Equivalent characterizations are: (i) cosθ=1, where θ is the angle between x and Ax. (ii) (x'Ax)^-1=xA^-1x. (iii) cosΦ=1, where φ is the angle between A^½x and A^-½x. We ask the question what is x such that cosθ 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 cosθ 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[(x'Ax)(x'A^-1x)]^-1 which gives the same anti-eigenvalue. The same result holds if the maximum of the angle Φ between A^½x and A^-½x 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.