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From: Stephen Montgomery-Smith on 16 Apr 2010 19:45
mateus.oliveira wrote:
> Let R(M) be the spectral radius of a matrix M and
> |M| be its Frobenius norm, i.e., the usual euclidean norm
> if we consider M as a vector.
>
> Is there any NxN matrix M such that
>
> |M|> N * R(M) ?
>
> Can someone come up with a simple example, or provide
> an argument that such a matrix cannot exist?
>
> cordially,
>
> mateus
>


Both the spectral radius and the Frobenius norm are unaffected by an
orthogonal change of basis. If M is symmetric, then it can be
diagonalized using an orthogonal change of basis.

Then the spectral radius is the maximum of the absolute value of the
eigenvalues, and the Frobenius norm is the square root of the sum of the
squares of the eigenvalues.

Hence |M| <= sqrt(N) R(M).


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