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Iterative Solution to a System of Matrix Equations

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  • Yong Lin
  • Qing-Wen Wang

Abstract

An efficient iterative algorithm is presented to solve a system of linear matrix equations A1X1B1 + A2X2B2 = E, C1X1D1 + C2X2D2 = F with real matrices X1 and X2. By this iterative algorithm, the solvability of the system can be determined automatically. When the system is consistent, for any initial matrices X10 and X20, a solution can be obtained in the absence of roundoff errors, and the least norm solution can be obtained by choosing a special kind of initial matrix. In addition, the unique optimal approximation solutions X∧1 and X∧2 to the given matrices X~1 and X~2 in Frobenius norm can be obtained by finding the least norm solution of a new pair of matrix equations A1X¯1B1+A2X¯2B2=E¯, C1X¯1D1+C2X¯2D2=F¯, where E¯=E-A1X~1B1-A2X~2B2, F¯=F-C1X~1D1-C2X~2D2. The given numerical example demonstrates that the iterative algorithm is efficient. Especially, when the numbers of the parameter matrices A1, A2, B1, B2, C1, C2, D1, D2 are large, our algorithm is efficient as well.

Suggested Citation

  • Yong Lin & Qing-Wen Wang, 2013. "Iterative Solution to a System of Matrix Equations," Abstract and Applied Analysis, John Wiley & Sons, vol. 2013(1).
    • Handle: RePEc:wly:jnlaaa:v:2013:y:2013:i:1:n:124979
    DOI: 10.1155/2013/124979
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    References listed on IDEAS

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    1. Ning Li & Qing-Wen Wang, 2013. "Iterative Algorithm for Solving a Class of Quaternion Matrix Equation over the Generalized (P, Q)‐Reflexive Matrices," Abstract and Applied Analysis, John Wiley & Sons, vol. 2013(1).
    2. Deqin Chen & Feng Yin & Guang-Xin Huang, 2012. "An Iterative Algorithm for the Generalized Reflexive Solution of the Matrix Equations AXB = E, CXD = F," Journal of Applied Mathematics, John Wiley & Sons, vol. 2012(1).
    3. Feng Yin & Guang-Xin Huang, 2012. "An Iterative Algorithm for the Least Squares Generalized Reflexive Solutions of the Matrix Equations AXB = E,CXD = F," Abstract and Applied Analysis, John Wiley & Sons, vol. 2012(1).
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    Cited by:

    1. Ivan I. Kyrchei, 2019. "Determinantal Representations of General and (Skew‐)Hermitian Solutions to the Generalized Sylvester‐Type Quaternion Matrix Equation," Abstract and Applied Analysis, John Wiley & Sons, vol. 2019(1).

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