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On Hermitian positive definite solutions of matrix equation X+A * X -2 A=I. (English) Zbl 1035.15017
The author considers the matrix equation X+A * X -2 A=I(1) and its Hermitian positive definite solutions; here A is an n×n complex matrix and I is the identity matrix of order n. He shows that if A is normal (i.e. AA * =A * A), then such a solution exists if and only if ρ(A)2/33 where ρ(A) is the spectral radius of A. The author discusses in detail the basic fixed point iterations for the equation in the case when A is nonnormal and A2/33 where · stands for the spectral norm for square matrices (i.e. one has 0A * A(4/27)I). Some of the results of I.G. Ivanov, V.I. Hasanov and B.V. Minchev [ibid. 326, 27-44 (2001; Zbl 0979.15007)] are improved.
MSC:
15A24Matrix equations and identities
65F10Iterative methods for linear systems
65F30Other matrix algorithms