TY - JOUR
T1 - A generalization of bergstrom's inequality and some applications
AU - Fong Yau, Sze
AU - Bresler, Yoram
N1 - Funding Information:
This work was supported in part by the National Science Foundation grant no. MIP-8810412, and the Joint Service Electronics Program grant no. NOOO14-90-J-1270. The Government has certain rights in this material.
PY - 1992/1/15
Y1 - 1992/1/15
N2 - Bergstrom's inequality is generalized. Using the new inequality, several interesting results concerning Hadamard product of matrices are proved. More specifically, let P,Q ε{lunate} CN×N such that P>0 and Q≥0, Qii>0. Then we prove the following tight inequalities: (a) Qii[(P⊙Q)-1]ii≤ (P-1ii; (b) Qii{[Re(P⊙Q)]-1}ii ≤ max {[Re(φPφ*)]-1}ii; (c) det(P⊙Q)≥ (detP)πNi=1 Qii; (d) det Re(P⊙ Q) ≥ min det[Re(φPφ*)]πNi=1 Qii, where in (b) and (d) maximization and minimization are over all unitary diagonal matrices φε{lunate}CN×N.
AB - Bergstrom's inequality is generalized. Using the new inequality, several interesting results concerning Hadamard product of matrices are proved. More specifically, let P,Q ε{lunate} CN×N such that P>0 and Q≥0, Qii>0. Then we prove the following tight inequalities: (a) Qii[(P⊙Q)-1]ii≤ (P-1ii; (b) Qii{[Re(P⊙Q)]-1}ii ≤ max {[Re(φPφ*)]-1}ii; (c) det(P⊙Q)≥ (detP)πNi=1 Qii; (d) det Re(P⊙ Q) ≥ min det[Re(φPφ*)]πNi=1 Qii, where in (b) and (d) maximization and minimization are over all unitary diagonal matrices φε{lunate}CN×N.
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U2 - 10.1016/0024-3795(92)90009-Y
DO - 10.1016/0024-3795(92)90009-Y
M3 - Article
AN - SCOPUS:44049116541
SN - 0024-3795
VL - 161
SP - 135
EP - 151
JO - Linear Algebra and Its Applications
JF - Linear Algebra and Its Applications
IS - C
ER -