On positive definite solutions of the family of matrix equations \(X+A^*X^{-n}A=Q\). (English) Zbl 1096.15003

Ivanov, Ivan G.

J. Comput. Appl. Math. 193, No. 1, 277-301 (2006).

The author studies the matrix equation \(X+A^*X^{-n}A=Q\) (properties of its maximal and minimal positive definite solutions) and the corresponding matrix function \[ G(X)=\root n\of{A(Q-X)^{-1}A^*}. \] He gives sufficient conditions for the existence of minimal and special positive definite solutions. The special positive definite solution \(X\) satisfies the condition \(\| X^{-1}\| \leq \frac{n+1}{n}\| Q^{-1}\| \). Iterative procedures for computing these solutions are discussed as well as conditions for convergence of the procedures.

Reviewer: Vladimir P. Kostov (Nice)

MSC:

15A24	Matrix equations and identities
15A45	Miscellaneous inequalities involving matrices
65F30	Other matrix algorithms (MSC2010)

Keywords:

nonlinear matrix equation; positive definite solution; iterative method; inequalities involving matrices; convergence

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