Abstract
For any symmetric orthogonal matrix P, i.e., \( P^{\rm T} = P,\,P^{\rm T} P = I, \) the matrix X is said to be a generalized centrosymmetric matrix if \( PXP = X \) for any matrix X. The conjugate gradient iteration algorithm is presented to find the generalized centrosymmetric solution and its optimal approximation of the constraint matrix equation \( AXB + CXD = F. \) By this method, the solvability of the equation can be determined automatically. If the matrix equation \( AXB + CXD = F \) is consistent, then its generalized centrosymmetric solution can be obtained within finite iteration steps in the absence of round off errors for any initial symmetric matrix \( X_{1} , \) and generalized centrosymmetric solution with the least norm can be derived by choosing a proper initial matrix. In addition, the optimal approximation solution for a given matrix of the matrix equation \( AXB + CXD = F \) can be obtained by choosing the generalized centrosymmetric solution with the least norm of a new matrix equation \( A\tilde{X}B + C\tilde{X}D = \tilde{F}. \)
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