... f1dm−1dm≤d1,(59)then the upper bound for the eigenvalue λ1(Q)givenby(34)in Theorem 11 is tighter than that by (2). The upper bounds for the eigenvalues λj(Q), j = 2, , m −1, provided by (34)in Theorem 11 ... thatλj(E)=dj(22) for j= 1, 2, , m.ByCorollary 2,wehaveλ1(S)= s, λm(S)= t, λj(S)= 0, for j = 2, , m −1,(23)where s and t are given by (21).Upper Bounds. By (14)inTheorem 6,wehaveλj(Q)≤ ... λm−1≤ dm−1≤ λm.(31)Proof. By using (19)withρ= 0inTheorem 8,wehaveλj≤ dj for j = 1, 2, , m − 1. By using (20)withρ = 1in Theorem 8,weobtainλj≥ dj−1 for j = 2, 3, , m.Combining these...