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Solving the nonlinear matrix equation X=Q+ i=1 m M i X δ i M i * via a contraction principle. (English) Zbl 1162.15008

Author’s abstract: “We consider the nonlinear matrix equation X=Q+ i=1 m M i X δ i M i * where Q is positive (resp. semidefinite) definite and the M i ’s are arbitrary (resp. nonsingular) matrices. We prove that if δ:=max{|δ i |:1im}<1, then the equation has a unique positive definite solution which is realized as the unique fixed point of a strict contraction with the Lipschitz constant less than or equal to δ. Furthermore, we show that the solution map varying over the determining coefficient matrices is continuous.”

The reader is recommended to read the following two papers to know previous results and methods related to the equation above: M.-J. Huang, C.-Y. Huang and T.-M. Tsai [ibid. 413, No. 1, 202–211 (2006; Zbl 1092.47053)] and X. Duan, A. Liao and B. Tang [ibid. 429, No. 1, 110–121 (2008; Zbl 1148.15012)].

15A24Matrix equations and identities
[1]Andruchow, E.; Corach, G.; Stojanoff, D.: Geometrical significance of lowner – Heinz inequality, Proc. amer. Math. soc. 128, 1031-1037 (2000) · Zbl 0945.46040 · doi:10.1090/S0002-9939-99-05085-6
[2]Bhatia, R.: On the exponential metric increasing property, Linear algebra appl. 375, 211-220 (2003) · Zbl 1052.15013 · doi:10.1016/S0024-3795(03)00647-5
[3]Corach, G.; Porta, H.; Recht, L.: Convexity of the geodesic distance on spaces of positive operators, Illinois J. Math. 38, 87-94 (1994) · Zbl 0802.53012
[4]Duan, X.; Liao, A.; Tang, B.: On the nonlinear matrix equation X-i=1mAi*xδiAi=Q, Linear algebra appl. 429, 110-121 (2008) · Zbl 1148.15012 · doi:10.1016/j.laa.2008.02.014
[5]Ferrante, A.; Levy, B.: Hermitian solutions of the equation X=Q+N*X-1N, Linear algebra appl. 247, 359-373 (1996) · Zbl 0876.15011 · doi:10.1016/0024-3795(95)00121-2
[6]Hasanov, V. I.: Positive definite solutions of the matrix equations X±ATX-qa=Q, Linear algebra appl. 404, 166-182 (2005) · Zbl 1078.15012 · doi:10.1016/j.laa.2005.02.024
[7]Heinz, E.: Beit age zur storungstheorie der spektralzerlegung, Math. ann. 123, 415-438 (1951) · Zbl 0043.32603 · doi:10.1007/BF02054965
[8]Huang, M.; Huang, C.; Tsai, T.: Applications of Hilbert’s projective metric to a class of positive nonlinear operators, Linear algebra appl. 413, 202-211 (2006) · Zbl 1092.47053 · doi:10.1016/j.laa.2005.08.024
[9]Lee, H.; Lim, Y.: Invariant metrics, contractions and nonlinear matrix equations, Nonlinearity 21, 857-878 (2008) · Zbl 1153.15020 · doi:10.1088/0951-7715/21/4/011
[10]Lawson, J.; Lim, Y.: Metric convexity of symmetric cones, Osaka J. Math. 44, 795-816 (2007) · Zbl 1135.53014 · doi:euclid:ojm/1199719405
[11]Liu, X. G.; Gao, H.: On the positive definite solutions of the matrix equation xs±ATX-ta=In, Linear algebra appl. 368, 83-97 (2003) · Zbl 1025.15018 · doi:10.1016/S0024-3795(02)00661-4
[12]Löwner, K.: Über monotone matrix functionen, Math. Z. 38, 177-216 (1934)
[13]Neeb, K. -H.: Compressions of infinite-dimensional bounded symmetric domains, Semigroup forum 61, 71-105 (2001) · Zbl 0980.22005 · doi:10.1007/s002330010037
[14]Nussbaum, R. D.: Hilbert’s projective metric and iterated nonlinear maps, Mem. amer. Math. soc. 391 (1988) · Zbl 0666.47028
[15]Thompson, A. C.: On certain contraction mappings in a partially ordered vector space, Proc. amer. Math. soc. 14, 438-443 (1963) · Zbl 0147.34903 · doi:10.2307/2033816