The geometric mean, matrices, metrics, and more

American Mathematical Monthly

Volumn 108, Issue 9, 2001, Pages 797-812

  Lawson, Jimmie D ^a   Lim, Yongdo ^b  

^a LOUISIANA STATE UNIVERSITY   (United States)

^b Kyungpook National University   (South Korea)

Author keywords

[No Author keywords available]

Indexed keywords

EID: 0035601362     PISSN: 00029890     EISSN: None     Source Type: Journal    
DOI: 10.2307/2695553     Document Type: Article

References (21)

1. W. N. Anderson, Jr., Shorted operators. SIAM J. Appl. Math. 20 (1971) 520-525.
2. T. Ando, Topics on Operator Inequalities, Lecture Notes Hokkaido Univ., Sapporo, 1978.
3. _, Concavity of certain maps on positive definite matrices and applications to Hadamard products, Linear Algebra Appl. 26 (1979) 203-241.
4. _, On some operator inequalities, Math. Ann. 279 (1987) 157-159.
5. G. Birkhoff. Extensions of Jentzsch's theorem, Trans. Amer. Math. Soc. 85 (1957) 219-227.
6. P. Bougerol, Kalman filtering with random coefficients and contractions, SIAM J. Control Optim. 31 (1993) 942-959.
7. F. Bruhat and J. Tits, Groupes Réductifs sur un Corps Local I, Pub. IHES 41 (1972) 5-251.
8. W. F. Donoghue, Jr., Monotone matrix functions and analytic continuation, Springer-Verlag, Berlin, 1974.
9. L. Faybusovich, Linear systems in Jordan algebras and primal-dual interior-point algorithms, J. Comput. Appl. Math. 86 (1997) 149-175.
10. _, Euclidean Jordan algebras and interior-point algorithms, Positivity 1 (1997) 331-357.
11. _, A Jordan-algebraic approach to potential-reduction algorithms, Technical Report, Department of Mathematics, University of Notre Dame, Notre Dame, 1998.
12. J. Faraut and A. Korányi, Analysis on symmetric cones, Clarendon Press, Oxford, 1994.
13. M. Fiedler and V. Pták, A new positive definite geometric mean of two positive definite matrices, Linear Algebra Appl. 251 (1997) 1-20.
14. R. Horn and C. Johnson, Matrix Analysis, Cambridge University Press, New York, 1985.
15. F. Kubo and T. Ando, Means of positive linear operators, Math. Ann. 246 (1980) 205-224.
16. S. Lang, Fundamentals of differential geometry, Graduate Texts in Math., Springer, Heidelberg, 1999.
17. Y. Lim, Geometric means on symmetric cones, Arch. Math. 75 (2000) 39-45.
18. H. Maass, Siegel's modular forms and Dirichlet series, Lecture Notes in Math. 216, Springer-Verlag, Heidelberg, 1971.
19. Y. E. Nesterov and A. S. Nemirovskii, Interior Point Polynomial Methods in Convex Programming, SIAM Publications, Philadelphia, 1993.
20. R. D. Nussbaum and J. E. Cohen, The arithmetic-geometric mean and its generalizations for noncommuting linear operators, Ann. Scuola. Norm. Sup. Pisa Cl. Sci. 15 (1988) 239-308.
21. G. Pedersen and M. Takesaki, The operator equation THT = K, Proc. Amer. Math. Soc. 36 (1972) 311-312.