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Iterative algorithms for solving a class of complex conjugate and transpose matrix equations. (English) Zbl 1222.65041

The authors consider a class of complex conjugate and transpose matrix equations

$\sum _{\ell =1}^{{s}_{1}}{A}_{\ell }X{B}_{\ell }+\sum _{\ell =1}^{{s}_{2}}{C}_{\ell }\overline{X}{D}_{\ell }+\sum _{\ell =1}^{{s}_{3}}{G}_{\ell }{X}^{T}{H}_{\ell }+\sum _{\ell =1}^{{s}_{4}}{M}_{\ell }{X}^{H}{N}_{\ell }=F$

in the unknown matrix $X\in {ℂ}^{r×s}$, which include equations of the form $AXB+CXD=F$, the normal Sylvester-conjugate matrix equations $AX-\overline{X}B=C$, $X-A\overline{X}B=C$, and the real matrix equation $AXB+C{X}^{T}D=F$. This kind of matrices are investigated to obtain a unified method for solving many complex matrix equations and to obtain insightful conclusions for some special matrices. By applying the hierarchical identification principle, an iterative algorithm is developed to solve such equations. With the aid of the real representation of a complex matrix, an easily computed sufficient condition is established to guarantee that the proposed algorithm is convergent for an arbitrary initial matrix in terms of the real representation of the coefficient matrices.

MSC:
 65F30 Other matrix algorithms 15A24 Matrix equations and identities 65F10 Iterative methods for linear systems