The Existence and Representation of the Solutions to the System of Operator Equations AiXBi + CiYDi + EiZFi = Gi(i = 1, 2)

Che, Gen; Hai, Guojun; Mei, Jiarui; Cao, **ang

doi:10.3390/axioms13070435

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The Existence and Representation of the Solutions to the System of Operator Equations A_iXB_i + C_iYD_i + E_iZF_i = G_i(i = 1, 2)

¹

School of Mathematical Sciences, Inner Mongolia University, Hohhot 010031, China

²

College of Science, Inner Mongolia Agricultural University, Hohhot 010031, China

^*

Author to whom correspondence should be addressed.

Axioms 2024, 13(7), 435; https://doi.org/10.3390/axioms13070435

Submission received: 21 April 2024 / Revised: 16 June 2024 / Accepted: 24 June 2024 / Published: 27 June 2024

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Abstract

In this paper, we give the necessary and sufficient conditions for the existence of general solutions, self-adjoint solutions, and positive solutions to the system of

A_{i} X B_{i} + C_{i} Y D_{i} + E_{i} Z F_{i} = G_{i} (i = 1, 2)

under additional conditions. In addition, we derive the representation of general solutions to the system of

A_{i} X B_{i} + C_{i} Y D_{i} + E_{i} Z F_{i} = G_{i} (i = 1, 2)

, and provide the matrix representation of the self-adjoint solutions and the positive solutions in the sense of the star order.

Keywords: operator equation; Moore–Penrose inverse; general solution; self-adjoint solution; positive solution

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MDPI and ACS Style

Che, G.; Hai, G.; Mei, J.; Cao, X. The Existence and Representation of the Solutions to the System of Operator Equations A_iXB_i + C_iYD_i + E_iZF_i = G_i(i = 1, 2). Axioms 2024, 13, 435. https://doi.org/10.3390/axioms13070435

AMA Style

Che G, Hai G, Mei J, Cao X. The Existence and Representation of the Solutions to the System of Operator Equations A_iXB_i + C_iYD_i + E_iZF_i = G_i(i = 1, 2). Axioms. 2024; 13(7):435. https://doi.org/10.3390/axioms13070435

Chicago/Turabian Style

Che, Gen, Guojun Hai, Jiarui Mei, and **ang Cao. 2024. "The Existence and Representation of the Solutions to the System of Operator Equations A_iXB_i + C_iYD_i + E_iZF_i = G_i(i = 1, 2)" Axioms 13, no. 7: 435. https://doi.org/10.3390/axioms13070435

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The Existence and Representation of the Solutions to the System of Operator Equations A_iXB_i + C_iYD_i + E_iZF_i = G_i(i = 1, 2)

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