Re: Bound for quotient of eigenvalues

In article <75dv9iF16ph6tU1@xxxxxxxxxxxxxxxxxx>,
José Carlos Santos <jcsantos@xxxxxxxx> wrote:

Hi all,

For each real-analytic function _f_ from [0,1] into itself, consider the
2x2 real matrix

a b

b c

where a = int(x^2 f(x),dx), b = int(x f(x),dx) and c= int(f(x),dx) (all
integrals are from 0 to 1). Unless _f_ is the null function, this matrix
is positive definite and therefore it has two positive eigenvalues.
Consider the quotient e_1/e_2 where e_1 is the greatest eigenvalue and
e_2 is the smallest one. My question is: is there an upper bound for
the quotients obtained by this method? My guess is that the answer is
negative, but I was unable to prove it.

Best regards,

Jose Carlos Santos

Take f_m(x) = (m+2)x^m. Then a = (m+2)/(m+3), b = 1, c = (m+2)/(m+1).
The matrix M_m -> the matrix of all 1's as m -> oo. The latter matrix
has eigenvalues 0 and 2. So you have to expect that for large m, the
eigenvalues of M_m are close to 0 and 2, and an easy argument shows
this is the case. It follows that e_1/e_2 is unbounded along this
sequence.
.

Relevant Pages