An iterative method for the least squares symmetric solution of the linear matrix equation AXB = C

This paper an iterative method is presented to solve the minimum Frobenius norm residual problem: min∥ AXB − C∥ with unknown symmetric matrix X. By this iterative method, for any initial symmetric matrix X 0, a solution X* can be obtained within finite iteration steps in the absence of roundoff erro...

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Bibliographic Details
Published in:Applied mathematics and computation 2005-11, Vol.170 (1), p.711-723
Main Author: Peng, Zhen-yun
Format: Article
Language:English
Subjects:
Algebra
Exact sciences and technology
Iterative method
Least-norm solution
Linear and multilinear algebra, matrix theory
Mathematics
Numerical analysis
Numerical analysis. Scientific computation
Numerical linear algebra
Sciences and techniques of general use
The matrix nearness problem
The minimum residual problem
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Summary:This paper an iterative method is presented to solve the minimum Frobenius norm residual problem: min∥ AXB − C∥ with unknown symmetric matrix X. By this iterative method, for any initial symmetric matrix X 0, a solution X* can be obtained within finite iteration steps in the absence of roundoff errors, and the solution X* with least norm can be obtained by choosing a special kind of initial symmetric matrix. In addition, the unique optimal approximation solution X ^ to a given matrix X ¯ in Frobenius norm can be obtained by first finding the least norm solution X ∼ ∗ of the new minimum residual problem: min ‖ A X ∼ B - C ∼ ‖ with unknown symmetric matrix X ∼ , where C ∼ = C - A X ¯ + X ¯ T 2 B . Given numerical examples are show that the iterative method is quite efficient.
ISSN:0096-3003
1873-5649
DOI:10.1016/j.amc.2004.12.032