Riccati equation and metric geometric means of positive semidefinite matrices involving semi-tensor products

Pattrawut Chansangiam; Arnon Ploymukda; Pattrawut Chansangiam; Arnon Ploymukda

doi:10.3934/math.20231195

AIMS Mathematics

2023, Volume 8, Issue 10: 23519-23533. doi: 10.3934/math.20231195

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Research article

Riccati equation and metric geometric means of positive semidefinite matrices involving semi-tensor products

Pattrawut Chansangiam ,
Arnon Ploymukda ^,

Department of Mathematics, School of Science, King Mongkut's Institute of Technology Ladkrabang, Bangkok 10520, Thailand

Received: 15 May 2023 Revised: 09 July 2023 Accepted: 17 July 2023 Published: 27 July 2023
MSC : 15A24, 15B48, 47A64

We investigate the Riccati matrix equation $ X A^{-1} X = B $ in which the conventional matrix products are generalized to the semi-tensor products $ \ltimes $. When $ A $ and $ B $ are positive definite matrices satisfying the factor-dimension condition, this equation has a unique positive definite solution, which is defined to be the metric geometric mean of $ A $ and $ B $. We show that this geometric mean is the maximum solution of the Riccati inequality. We then extend the notion of the metric geometric mean to positive semidefinite matrices by a continuity argument and investigate its algebraic properties, order properties and analytic properties. Moreover, we establish some equations and inequalities of metric geometric means for matrices involving cancellability, positive linear map and concavity. Our results generalize the conventional metric geometric means of matrices.
- metric geometric mean,
- semi-tensor product,
- positive definite matrix,
- Riccati equation
Citation: Pattrawut Chansangiam, Arnon Ploymukda. Riccati equation and metric geometric means of positive semidefinite matrices involving semi-tensor products[J]. AIMS Mathematics, 2023, 8(10): 23519-23533. doi: 10.3934/math.20231195

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Abstract

We investigate the Riccati matrix equation $ X A^{-1} X = B $ in which the conventional matrix products are generalized to the semi-tensor products $ \ltimes $. When $ A $ and $ B $ are positive definite matrices satisfying the factor-dimension condition, this equation has a unique positive definite solution, which is defined to be the metric geometric mean of $ A $ and $ B $. We show that this geometric mean is the maximum solution of the Riccati inequality. We then extend the notion of the metric geometric mean to positive semidefinite matrices by a continuity argument and investigate its algebraic properties, order properties and analytic properties. Moreover, we establish some equations and inequalities of metric geometric means for matrices involving cancellability, positive linear map and concavity. Our results generalize the conventional metric geometric means of matrices.

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