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On the matrix equation \(X-A^* X^{-n} A = I\). (English) Zbl 1081.65036

The authors consider the nonlinear matrix equation \[ X-A^{\star}X^{-n}A=I, \] where \(X\) is an unknown matrix, \(I\) is the \(m\times m\) identity matrix and \(n\) is a positive integer. The equations \(X+A^{\star}X^{-1}A=Q\) and \(X-A^{\star}X^{-1}A=Q\) have many applications, and iterative procedures for solving the equation \(X-A^{\star}X^{-1}A=Q\) have been proposed [see C.-H. Guo and P. Lancaster, Math. Comput. 68, 1589–1603 (1999; Zbl 0940.65036) and A. Ferrante and B. C. Levy, Linear Algebra Appl. 247, 359–373 (1996; Zbl 0876.15011 )]. The iterative positive definite solutions and the properties of the equations \(X-A^{\star}X^{-2}A=I,\) and \(X+A^{\star}X^{-2}A=I\) have been discussed by I. G. Ivanov and S. M. El-Sayed [Linear Algebra Appl. 279, 303–316 (1998; Zbl 0935.65041)], and by I. G. Ivanov, V. I. Hasanov and B. V. Minchov [Linear Algebra Appl. 326, No. 1–3, 27–44 (2001; Zbl 0979.15007)].
In this paper, the authors review the existing methods for solving the equation \(X-A^{\star}X^{-n}A=I,\) and derive a sufficient condition for this equation to have a unique positive definite solution. Moreover, the convergence of the iterative methods proposed by S. M. El-Sayed [Comput. Math. Appl. 41, 579–588 (2001; Zbl 0984.65043)] is proved under weaker restrictions for the matrix \(A\).

MSC:

65F30 Other matrix algorithms (MSC2010)
15A24 Matrix equations and identities
65F10 Iterative numerical methods for linear systems
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References:

[1] Bhatia, R., Matrix Analysis, Graduate Text in Mathematics, vol. 169 (1997), Springer-Verlag
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