上海大学学报(自然科学版) ›› 2014, Vol. 20 ›› Issue (2): 138-156.doi: 10.3969/j.issn.1007-2861.2014.01.021

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A Pair of Mixed Generalized Sylvester Matrix Equations

HE Zhuo-heng, WANG Qing-wen   

  1. (College of Sciences, Shanghai University, Shanghai 200444, China)
  • 出版日期:2014-04-26 发布日期:2014-04-26
  • 通讯作者: Wang Qing-wen, Professor, Ph. D., E-mail: wqw369@yahoo.com
  • 基金资助:

    the National Natural Science Foundation of China (11171205), and the Key Project of the Scientific Research Innovation Foundation of Shanghai Municipal Education Comm-ission (13ZZ080)

A Pair of Mixed Generalized Sylvester Matrix Equations

HE Zhuo-heng, WANG Qing-wen   

  1. (College of Sciences, Shanghai University, Shanghai 200444, China)
  • Online:2014-04-26 Published:2014-04-26
  • Contact: Wang Qing-wen, Professor, Ph. D., E-mail: wqw369@yahoo.com
  • Supported by:

    the National Natural Science Foundation of China (11171205), and the Key Project of the Scientific Research Innovation Foundation of Shanghai Municipal Education Comm-ission (13ZZ080)

摘要: In this paper, some necessary and sufficient solvability conditions for the system of mixed generalized Sylvester matrix equations A1X − Y B1 = C1, A2Y − ZB2 = C2 are derived, and an expression of the general solution to this system is given when it is solvable. Admissible ranks of the solution, and admissible ranks and inertias of the Hermitian part of the solution are investigated, respectively. As an application of the above system, solvability conditions and the general Hermitian solution to the generalized Sylvester matrix equation are obtained. Moreover, we provide an algorithm and an example to illustrate our results.

关键词: explicit solutions, inertia, Moore-Penrose inverse, rank, Sylvester equation

Abstract: In this paper, some necessary and sufficient solvability conditions for the system of mixed generalized Sylvester matrix equations A1X − Y B1 = C1, A2Y − ZB2 = C2 are derived, and an expression of the general solution to this system is given when it is solvable. Admissible ranks of the solution, and admissible ranks and inertias of the Hermitian part of the solution are investigated, respectively. As an application of the above system, solvability conditions and the general Hermitian solution to the generalized Sylvester matrix equation are obtained. Moreover, we provide an algorithm and an example to illustrate our results.

Key words: explicit solutions, inertia, Moore-Penrose inverse, rank, Sylvester equation

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