The generalized bisymmetric solutions of the matrix equation A 1 X 1 B 1 + A 2 X 2 B 2 +  ⋯ + A l  X l  B l = C and its optimal approximation

Abstract

For fixed generalized reflection matrix P, i.e. P T = P, P 2 = I, then matrix X is said to be generalized bisymmetric, if X = X T = PX P. In this paper, an iterative method is constructed to find the generalized bisymmetric solutions of the matrix equation A 1 X 1 B 1 + A 2 X 2 B 2 + · · · + A l X l B l = C where [X 1 , X 2 , · · · , X l ] is real matrices group. By this iterative method, the solvability of the matrix equation can be judged automatically. When the matrix equation is consistent, for any initial generalized bisymmetric matrix group X (0) 1 , X (0) 2 , · · · , X (0) l , a generalized bisymmetric solution group can be obtained within finite iteration steps in the absence of roundoff errors, and the least norm generalized bisymmetric solution group can be obtained by choosing a special kind of initial generalized bisymmetric matrix group. In addition, the optimal approximation generalized bisymmetric solution group to a given generalized bisymmetric matrix group X 1 , X 2 , · · · , X l in Frobenius norm can be obtained by finding the least norm generalized bisymmetric solution group of the new matrix equation A 1 X 1 B 1 + A 2 X 2 B 2 + · · · + A l X l B l = C, where C = C − A 1 X 1 B 1 − A 2 X 2 B 2 − · · · − A l X l B l . Given numerical examples show that the algorithm is efficient.

CO2-in-PEG emulsion-templating synthesis of poly(acrylamide) with controllable porosity and their use as efficient catalyst supports

“…30 The excess macropores with sizes in the hundreds of nanometers range as shown in Fig. This could be resulted from that CO 2 could be solubilized in bulk PEG-400 except for dissolving inside micelles and the amount of CO 2 solubilized in the bulk phase increased with CO 2 pressure, a phenomenon which had been reported for CO 2 -in-water micelles, 43,44 and thus caused the PAM synthesized at higher pressure was more porous after releasing CO 2 . In addition, the effect of the CO 2 was also more obvious on the distribution of the macropores (Fig.…”

An Iterative Algorithm for the Least Squares Solutions of Matrix Equations Over Symmetric Arrowhead Matrices

“…Hitherto, several types of iterative algorithms to solve various kinds of matrix equations have been handled in the literature; for further details see [1,2,10,11,12,15,18,19,26,29,30,34,35,37,38,43] and the references therein. In addition the explicit forms of the solutions of some kinds of matrix equations have been derived.…”

“…The idea of conjugate gradient (CG) method [33] has been developed for constructing iterative algorithms to compute the solutions of different kinds of linear matrix equations over generalized reflexive and anti-reflexive, generalized bisymmetric, generalized centro-symmetric, mirror-symmetric, skewsymmetric and (P, Q)-reflexive matrices, for more details see [2,7,8,9,16,22,29,38] and the references therein. For instance, Peng et al [28] have proposed an iterative algorithm for finding the bisymmetric solutions of matrix equation…”

Solving coupled matrix equations over generalized bisymmetric matrices

“…For example, Dehghan and Hajarian [19][20][21][22][23][24][25][26] proposed several efficient iterative algorithms for solving Sylvester matrix equations. In [27], an efficient method was proposed to find the generalized bisymmetric solutions of the matrix equation In [28], Cramer's rules for some quaternion matrix equations were presented within the framework of the theory of the column and row determinants. Kyrchei [29] considered systems of linear quaternionic equations and obtained Cramer's rules for right and left quaternionic systems of linear equations.…”