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.