From: borchers@nmt.edu (Brian Borchers)
Subject: Re: eigen values and norms?
Date: 21 Apr 1999 15:51:04 GMT
Newsgroups: sci.math.num-analysis
Keywords: matrix norms match largest eigenvalue

Henry Wolkowicz (hwolkowi@orion.math.uwaterloo.ca) wrote
>
>In fact, there is one additional fact. There are many matrix norms, e.g.
>the ones that are induced by different vector norms. Then the largest
>eigenvalue (spectral radius in fact) is the inf of all these matrix norms.
>I cannot remember the reference for this theorem at the moment. 
>Also, I think the inf is over 'vector norm induced' matrix norms but am not
>certain.

See Householder, "The Theory of Matrices in Numerical Analysis", section
2.3.  For any matrix A and any epsilon>0, there is a vector norm K such
that the matrix norm K induced by the vector norm K is such that 
 
  || A ||   <= epsilon + rho(A)
         K

where rho(A) is the maximum of the absolute values of the (complex)
eigenvalues of A.  (rho(A) is known as the spectral radius of A.)
Thus rho(A) is the infinum of the vector norm induced matrix norms of
A.

Note that Householder writes all of this in rather unusual notation-
you'll need to go back through the chapter to translate it into the
conventional notation used above.
 
With respect to the standard L_1, L_2, and L_infinity norms, all that
you can say is that rho(A) <= || A ||.

-- 
Brian Borchers                              borchers@nmt.edu
Department of Mathematics                   http://www.nmt.edu/~borchers/
New Mexico Tech                             Phone: 505-835-5813
Socorro, NM 87801                           FAX: 505-835-5366