LOWER BOUNDS FOR THE NUMERICAL RADIUS

Abstract

C We show that if A = [a(ij)](i)(n),(j=1) is an n-by-n complex matrix and A '= [a '(ij)](i)(n),(j=1), where a '(ij) ={a(ij) if (i,j) = (1,2),..., (n-1,n) or (n,1), 0 otherwise, then w(A) >= w(A '), where w((.)) denotes the numerical radius of a matrix. Moreover, if n is odd and a(12),..., a(n-1),(n),a(n1) are all nonzero, then w(A) = w(A ') if and only if A = A '. For an even n, under the same nonzero assumption, we have W(A) = W(A ') if and only if A = A ', where W((.)) is the numerical range of a matrix.