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Positive definite solution of the matrix equation \(\boldsymbol {X=Q+A^{H}(I\otimes X-C)^{\delta}A}\)

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Abstract

Consider the nonlinear matrix equation X = Q + A H(I ⊗ X − C)δ A ( δ = − 1 or 0 < |δ| < 1), where Q is an n×n positive definite matrix, C is an mn ×mn positive semidefinite matrix, I is an m×m identity matrix, and A is an arbitrary mn×n matrix. This equation is connected with a certain interpolation problem when δ = − 1. Using the properties of the Kronecker product and the theory for the monotonic operator defined in a normal cone, we prove the existence and uniqueness of the positive definite solution which is contained in the set {X|I ⊗ X > C} under the condition that I ⊗ Q > C. The iterative methods to compute the unique solution is proposed. Numerical examples show that the methods are feasible and effective.

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Authors and Affiliations

  1. College of Mathematics and Computing Science, Changsha University of Science and Technology, Changsha, 410114, People’s Republic of China

    Guozhu Yao

  2. College of Mathematics and Econometrics, Hunan University, Changsha, 410082, People’s Republic of China

    Guozhu Yao & Anping Liao

  3. School of Mathematics and Computing Science, Guilin University of Electronic Technology, Guilin, 541004, People’s Republic of China

    Xuefeng Duan

Authors
  1. Guozhu Yao
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  2. Anping Liao
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  3. Xuefeng Duan
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Correspondence to Guozhu Yao.

Additional information

The work was supported in part by Natural Science Foundation of Hunan Province (09JJ6012).

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Yao, G., Liao, A. & Duan, X. Positive definite solution of the matrix equation \(\boldsymbol {X=Q+A^{H}(I\otimes X-C)^{\delta}A}\) . Numer Algor 56, 349–361 (2011). https://doi.org/10.1007/s11075-010-9386-9

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  • DOI: https://doi.org/10.1007/s11075-010-9386-9

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