Metric geometric means with arbitrary weights of positive definite matrices involving semi-tensor products

Arnon Ploymukda; Pattrawut Chansangiam; Arnon Ploymukda; Pattrawut Chansangiam

doi:10.3934/math.20231333

AIMS Mathematics

2023, Volume 8, Issue 11: 26153-26167. doi: 10.3934/math.20231333

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Metric geometric means with arbitrary weights of positive definite matrices involving semi-tensor products

Arnon Ploymukda ,
Pattrawut Chansangiam ^,

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

Received: 04 August 2023 Revised: 31 August 2023 Accepted: 04 September 2023 Published: 12 September 2023
MSC : 15A24, 15A60, 15A69

We extend the notion of classical metric geometric mean (MGM) for positive definite matrices of the same dimension to those of arbitrary dimensions, so that usual matrix products are replaced by semi-tensor products. When the weights are arbitrary real numbers, the weighted MGMs possess not only nice properties as in the classical case, but also affine change of parameters, exponential law, and cancellability. Moreover, when the weights belong to the unit interval, the weighted MGM has remarkable properties, namely, monotonicity and continuity from above. Then we apply a continuity argument to extend the weighted MGM to positive semidefinite matrices, here the weights belong to the unit interval. It turns out that this matrix mean posses rich algebraic, order, and analytic properties, such as, monotonicity, continuity from above, congruent invariance, permutation invariance, affine change of parameters, and exponential law. Furthermore, we investigate certain equations concerning weighted MGMs of positive definite matrices. It turns out that such equations are always uniquely solvable with explicit solutions. The notion of MGMs can be applied to solve certain symmetric word equations in two letters.
- weighted metric geometric mean,
- positive definite matrix,
- semi-tensor product,
- symmetric word equation
Citation: Arnon Ploymukda, Pattrawut Chansangiam. Metric geometric means with arbitrary weights of positive definite matrices involving semi-tensor products[J]. AIMS Mathematics, 2023, 8(11): 26153-26167. doi: 10.3934/math.20231333

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Abstract

We extend the notion of classical metric geometric mean (MGM) for positive definite matrices of the same dimension to those of arbitrary dimensions, so that usual matrix products are replaced by semi-tensor products. When the weights are arbitrary real numbers, the weighted MGMs possess not only nice properties as in the classical case, but also affine change of parameters, exponential law, and cancellability. Moreover, when the weights belong to the unit interval, the weighted MGM has remarkable properties, namely, monotonicity and continuity from above. Then we apply a continuity argument to extend the weighted MGM to positive semidefinite matrices, here the weights belong to the unit interval. It turns out that this matrix mean posses rich algebraic, order, and analytic properties, such as, monotonicity, continuity from above, congruent invariance, permutation invariance, affine change of parameters, and exponential law. Furthermore, we investigate certain equations concerning weighted MGMs of positive definite matrices. It turns out that such equations are always uniquely solvable with explicit solutions. The notion of MGMs can be applied to solve certain symmetric word equations in two letters.

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