Re: eigenvalue magnitude question


In article <uCRPf.35282$Ru5.34795@xxxxxxxxxxxxxxxxxxxx>,
"NoIdea" <foru@xxxxxxxxx> writes:
would the fact that the highest singular value of a real/complex matrix is
less than 1 , necessarily imply that all of its eigenvalues have magnitudes
less than 1 ?

thanks



take
A*x=lambda*x and w.l.o.g. ||x||_2=1
insert
A=U*S*V' U,V unitary
multiply by U'
S*V'*x= lambda*U'*x
take the 2-norm on both sides and use submultiplicativity
max|S(i,i)| ||V'*x||_2 >= abs(lambda)||U'*x||_2
but ||x||_2=1 and U', V' unitary
done

hth
peter
.

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