Solvability of the matrix equation $ AX^{2} = B $ with semi-tensor product

Jin Wang; Jun-E Feng; Hua-Lin Huang; Jin Wang; Jun-E Feng; Hua-Lin Huang

doi:10.3934/era.2020114

Electronic Research Archive

2021, Volume 29, Issue 3: 2249-2267. doi: 10.3934/era.2020114

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Solvability of the matrix equation $ AX^{2} = B $ with semi-tensor product

Jin Wang ^{1
,
,},
Jun-E Feng ^{1,2
,},
Hua-Lin Huang ^{1,2
,}

1.
School of Mathematics, Shandong University, Jinan 250100, China
2.
School of Mathematical Sciences, Huaqiao University, Quanzhou 362021, China

Received: 01 May 2020 Revised: 01 August 2020 Published: 19 October 2020
Primary: 15A06; Secondary: 15A03, 15A04

We investigate the solvability of the matrix equation $ AX^{2} = B $ in which the multiplication is the semi-tensor product. Then compatible conditions on the matrices $ A $ and $ B $ are established in each case and necessary and sufficient condition for the solvability is discussed. In addition, concrete methods of solving the equation are provided.
- Matrix equation,
- matrix-vector equation,
- semi-tensor product,
- high ordered matrix equation,
- Toeplitz matrix
Citation: Jin Wang, Jun-E Feng, Hua-Lin Huang. Solvability of the matrix equation $ AX^{2} = B $ with semi-tensor product[J]. Electronic Research Archive, 2021, 29(3): 2249-2267. doi: 10.3934/era.2020114

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Abstract

We investigate the solvability of the matrix equation $ AX^{2} = B $ in which the multiplication is the semi-tensor product. Then compatible conditions on the matrices $ A $ and $ B $ are established in each case and necessary and sufficient condition for the solvability is discussed. In addition, concrete methods of solving the equation are provided.

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