EIGENVALUE LOCALIZATION FOR COMPLEX MATRICES

Let A be an n x n complex matrix with n >= 3. It is shown that at least n - 2 of the eigenvalues of A lie in the disk vertical bar z - tr A/n vertical bar <= root n-1/n(root(parallel to A parallel to(2)(2) - vertical bar trA vertical bar(2)/n)(2) - parallel to A*A-AA*parallel to(2)(2) - spd(2)(A)/2), where parallel to A parallel to(2) , tr A, and spd(A) denote the Frobenius norm, the trace, and the spread of A, respectively. In particular, if A = [a(ij)] is normal, then at least n - 2 of the eigenvalues of A lie in the disk vertical bar z - tr A/n vertical bar <=root n - 1/n(parallel to A parallel to(2)(2)/2 - vertical bar tr A vertical bar(2)/n - 3/2 (i,j-1,...,n)max (Sigma(k=1) (n)(k not equal i)vertical bar a(ki)vertical bar(2) + Sigma(k=1) (n)(k not equal j) vertical bar a(kj)vertical bar(2) + vertical bar a(ii) - a(jj)vertical bar(2))). Moreover, the constant 3/2 can be replaced by 4 if the matrix A is Hermitian.