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Generalized product of two square matrices and application for some algebraic equations. (English) Zbl 1429.15020
Authors’ abstract: In this paper, a generalized matrix product is introduced and related properties are studied as well. Afterwards, we show how our approach can be applied to the so-called Sylvester and Lyaponov matrix equations for obtaining their related solutions in terms of the generalized matrix product. Numerical examples illustrating the theoretical study are also discussed.
MSC:
15A60 Norms of matrices, numerical range, applications of functional analysis to matrix theory
15A24 Matrix equations and identities
26A51 Convexity of real functions in one variable, generalizations
47A64 Operator means involving linear operators, shorted linear operators, etc.
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[1] Ando, T; Li, CK; Mathias, R, Geometric means, Linear Algebra Appl., 38, 305-334, (2004) · Zbl 1063.47013
[2] Atteia, M; Raïssouli, M, Self dual operators on convex functionals, geometric mean and square root of convex functionals, J Convex Anal, 8, 223-240, (2001) · Zbl 1003.90030
[3] Bhatia, R.: Matrix analysis. Verlag, New York (1997) · Zbl 0863.15001
[4] Bhatia, R; Uchiyama, M, The operator equation \(∑ _{i=1}^nA^{n-1}XB^i=Y\), Expo. Math., 27, 251-255, (2009) · Zbl 1167.15010
[5] Chan, NN; Kwong, MK, Hermitian matrix inequalities and a conjecture, Am. Math. Monthly, 92, 533-541, (1985) · Zbl 0587.15009
[6] Cardoso, JR, Computation of the pth root and its Fréchet derivative by integrals, Electr. Trans. Num. An., 39, 414-436, (2012) · Zbl 1287.65035
[7] Furuta, T, Positive sem-definite solution of the operator equation \(∑ _{j=1}^nA^{n-j}XA^{j-1}=B\), Linear Algebra Appl., 432, 949-955, (2009) · Zbl 1187.15017
[8] Kato, T.: Perturbation theory for linear operators. Springer, Berlin (1984) · Zbl 0531.47014
[9] Lefschetz, S., Lasalle, J.P.: Stability by Lyapunov direct method. Academic Press, New York (1987)
[10] Raïssouli, M; Leazizi, F, Continued fraction expansion of the geometric matrix mean and applications, Linear Algebra Appl., 359, 37-57, (2003) · Zbl 1027.15018
[11] Raïssouli, M; Bouziane, H, Arithmetico-geometrico-harmonic functional mean in the sense of convex analysis, Ann. Sci. Math. Québec, 30, 79-107, (2006) · Zbl 1222.49025
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