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Nonnegative definite and Re-nonnegative definite solutions to a system of matrix equations with statistical applications

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  • Song, Guangjing
  • Yu, Shaowen

Abstract

Necessary and sufficient conditions are given for the existence of a nonnegative definite solution, a Re-nonnegative definite solution, a positive definite solution and a Re-positive definite solution to the system of matrix equations AXA*=CandBXB*=D,respectively. The expressions for these special solutions are given when the consistent conditions are satisfied. Based on the new results, the characterization of the covariance matrix such that a pair of multivariate quadratic forms are distributed as independent noncentral Wishart random matrices is derived. Many results existing in the literature are extended.

Suggested Citation

  • Song, Guangjing & Yu, Shaowen, 2018. "Nonnegative definite and Re-nonnegative definite solutions to a system of matrix equations with statistical applications," Applied Mathematics and Computation, Elsevier, vol. 338(C), pages 828-841.
    • Handle: RePEc:eee:apmaco:v:338:y:2018:i:c:p:828-841
    DOI: 10.1016/j.amc.2018.06.045
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    References listed on IDEAS

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    1. Tongsong Jiang & Xuehan Cheng & Sitao Ling, 2014. "An Algebraic Relation between Consimilarity and Similarity of Quaternion Matrices and Applications," Journal of Applied Mathematics, Hindawi, vol. 2014, pages 1-5, September.
    2. Young, Dean M. & Seaman, John W. & Meaux, Laurie M., 1999. "Independence Distribution Preserving Covariance Structures for the Multivariate Linear Model," Journal of Multivariate Analysis, Elsevier, vol. 68(2), pages 165-175, February.
    3. Yonghui Liu & Yongge Tian, 2011. "Max-Min Problems on the Ranks and Inertias of the Matrix Expressions A−BXC±(BXC)∗ with Applications," Journal of Optimization Theory and Applications, Springer, vol. 148(3), pages 593-622, March.
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