... m.(45)Proof. In Theorem 8, the lower bounds of the eigenvalues of Q in (20) are determined bydj, j = 1,2, , m,ands and t in (21). As we did in Theorem 12, the lower bounds of the eigenvalues of ... m.(34)Proof. In Theorem 8, the upper bounds of the eigenvalues of Q in (19) are determined bydj, j = 1,2, ,m,ands and t in (21). They can be viewed as functions of ρ in [0, 1]. Thatis, the upper ... toreexamines these bounds of the eigenvalues of Q and makesthem sharper. In [9], the hub matrix theory is also appliedto the MIMO beamforming problem by comparing k of mtransmitting antennas...