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An iteration method for the symmetric solutions and the optimal approximation solution of the matrix equation AXB=C. (English) Zbl 1068.65056

Let m×n be the set of all m×n matrices, S n the set of all symmetric matrices in n×n . For A m×n , A denotes the Frobenius norm. The authors consider the following two problems. Problem 1. Given A m×n , B n×p , C m×p , find XS n such that AXB=C. Problem 2. If Problem 1 is consistent, then denote its solutions by 𝒮 E . For given X 0 n×n , find X ^𝒮 E such that

X ^-X 0 =min{X-X 0 :X𝒮 E }·

The authors describe an iterative method that determines the solvability of Problem 1 automatically and in the case of solvability computes a solution in an a priori known finite number of steps. Furthermore, the solution to Problem 2 can be found by choosing a suitable initial iteration matrix. It can also be found as the least-norm solution to another equation AX ¯B=C ¯. The paper is carefully written with detailed and convincing proofs. It also contains a numerical example.

65F30Other matrix algorithms
65F10Iterative methods for linear systems
15A24Matrix equations and identities