From: ilya@math.ohio-state.edu (Ilya Zakharevich)
Newsgroups: sci.math.research
Subject: Re: Kantorovich's Inequality Question
Date: 23 Nov 1998 00:51:00 GMT
[A complimentary Cc of this posting was sent to Scott D. Berger
],
who wrote in article <36578368.2039DE0A@afit.af.mil>:
> Let U be a positive definite Hermitian 2n x 2n matrix and z be a 2n x 1
> vector. Kantorovich's equality states that
>
> (z^H U^-1 z)(z^H U z) <= ( (a+b)^2 / (4*a*b) ) (z^H z)^2
>
> where z^H denote the complex conjugate transpose of z, U^-1 denotes the
> inverse of U, <= denotes less than or equal to, a denotes the largest
> eigenvalues of U, and b denotes the smallest eigenvalue of U.
[...]
> With this additional information about the eigenvalues of U can we
> improve upon Kantorovich's inequality? Or can we only improve upon
> Kantorovich's inequality with additional information about z?
If you allow z to vary arbitrary, the above inequality is (obviously)
sharp (restrict it to 2-dim subspace spanned by 2 eigenvectors with
max and min eigenvalue). So the only way to improve it is to control
z.
Note that both in the original formulation and in the modified one one
can assume all the variable matrices to be real diagonal with positive
diagonal elements. Maybe this remark will help you to understand what
you actually wanted to ask. ;-)
Ilya