Solvability of the Sylvester equation $ AX-XB = C $ under left semi-tensor product

Naiwen Wang; Naiwen Wang

doi:10.3934/mmc.2022010

Mathematical Modelling and Control

2022, Volume 2, Issue 2: 81-89. doi: 10.3934/mmc.2022010

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Research article

Solvability of the Sylvester equation $ AX-XB = C $ under left semi-tensor product

Naiwen Wang ^,

School of Mathematics and Statistics, Shandong Normal University, Jinan, 250014, China

Received: 31 March 2022 Revised: 08 April 2022 Accepted: 20 April 2022 Published: 27 June 2022

This paper investigates the solvability of the Sylvester matrix equation $ AX-XB = C $ with respect to left semi-tensor product. Firstly, we discuss the matrix-vector equation $ AX-XB = C $ under semi-tensor product. A necessary and sufficient condition for the solvability of the matrix-vector equation and specific solving methods are studied and given. Based on this, the solvability of the matrix equation $ AX-XB = C $ under left semi-tensor product is discussed. Finally, several examples are presented to illustrate the efficiency of the results.
- Sylvester matrix equation,
- left semi-tensor product,
- solvability,
- matrix-vector equation,
- necessary and sufficient condition
Citation: Naiwen Wang. Solvability of the Sylvester equation $ AX-XB = C $ under left semi-tensor product[J]. Mathematical Modelling and Control, 2022, 2(2): 81-89. doi: 10.3934/mmc.2022010

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Abstract

This paper investigates the solvability of the Sylvester matrix equation $ AX-XB = C $ with respect to left semi-tensor product. Firstly, we discuss the matrix-vector equation $ AX-XB = C $ under semi-tensor product. A necessary and sufficient condition for the solvability of the matrix-vector equation and specific solving methods are studied and given. Based on this, the solvability of the matrix equation $ AX-XB = C $ under left semi-tensor product is discussed. Finally, several examples are presented to illustrate the efficiency of the results.

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