Re: bounds on Raleigh quotient

Simon wrote:
>
> > Hi,
> > your computations are incorrect. You forget to take the nonnegativity
> > constraint into account.
> > Take as an example the matrix a11=a22=2, a12=a21=1. It's smallest
> > eigenvalue is 1. The smallest value of the Rayleigh quotient for x>=0
> > is at least 2.
> > Hans Mittelmann
>
> Can you please point out how this example makes my computations
> incorrect? I stated in my orignal post that the minimum
> eigenvalue is a lower bound on the minimum Raleigh quotient
> subject to nonnegativity of x, and your example verifies that.
>
> Simon

Although the eigenvalues of the given matrix are 1 and 3, if you
calculate the Rayleigh quotient using a vector with entirely
positive components, you are going to get at least 2. I think
that is what Herr Professor Mittelmann means.

--
Julian V. Noble
Professor Emeritus of Physics
jvn@xxxxxxxxxxxxxxxxxxxxxxxxxxxxxx
^^^^^^^^^^^^^^^^^^
http://galileo.phys.virginia.edu/~jvn/

"For there was never yet philosopher that could endure the
toothache patiently."

-- Wm. Shakespeare, Much Ado about Nothing. Act v. Sc. 1.
.

Relevant Pages