Chansangiam, Pattrawut

Preferred name

Alternative Name

Chansangiam, P.

Main Affiliation

School of Science

pattrawut.ch@kmitl.ac.th

Now showing 1 - 6 of 6

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Least-squares solutions of generalized linear systems and the matrix equation AXB = C under the general semi-tensor products
(2026-01-01)
Jaiprasert, Janthip
;
Phoonphiphat, Thanaphon
;
Chansangiam, Pattrawut
;
Zhang, Yang
We investigate least-squares (LS) solutions of Sylvester-type matrix equations formulated via the general semi-tensor product (GSTP) of matrices. In particular, we consider generalized linear systems of the form A (Formula presented) x = B, where A and B are given rectangular matrices and x is an unknown column vector, with k denoting the GSTP that extends both the conventional matrix product and the semi-tensor product. By analyzing the derivative of the LS error associated with the equation, we show that LS solutions can be obtained by solving an equivalent linear system under the usual matrix product. Using matrix partitioning techniques, these results are further extended to several Sylvester-type equations, including A (Formula presented) X = B, X (Formula presented) A = B. and A (Formula presented) X x B = C, where X is an unknown matrix of compatible size. This framework unifies the classical and semi-tensor product cases under a generalized algebraic setting. Furthermore, we develop a gradient-descent iterative algorithm to compute approximate LS solutions efficiently. Numerical experiments confirm the convergence, capability, and effectiveness of the proposed method.
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Riccati equation and metric geometric means of positive semidefinite matrices involving semi-tensor products
(2023-01-01)
Chansangiam, Pattrawut
;
Ploymukda, Arnon
We investigate the Riccati matrix equation XA−1X = B in which the conventional matrix products are generalized to the semi-tensor products ⋉. 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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Solving the Sylvester-Transpose Matrix Equation under the Semi-Tensor Product
(2022-06-01)
Jaiprasert, Janthip
;
Chansangiam, Pattrawut
This paper investigates the Sylvester-transpose matrix equation A ⋉ X + XT ⋉ B = C, where all mentioned matrices are over an arbitrary field. Here, ⋉ is the semi-tensor product, which is a generalization of the usual matrix product defined for matrices of arbitrary dimensions. For matrices of compatible dimensions, we investigate criteria for the equation to have a solution, a unique solution, or infinitely many solutions. These conditions rely on ranks and linear dependence. Moreover, we find suitable matrix partitions so that the matrix equation can be transformed into a linear system involving the usual matrix product. Our work includes the studies of the equation A ⋉ X = C, the equation X ⋉ B = C, and the classical Sylvester-transpose matrix equation.
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Sylvester matrix equation under the semi-tensor product of matrices
(2022-01-01)
Chansangiam, Pattrawut
;
Sabau, Sorin V.
We investigate the Sylvester matrix equation in which the product is given by the semi-tensor product, and all involved matrices are matrices over an arbitrary field. We discuss necessary/sufficient condition(s) for the matrix equation to have a solution or a unique solution, or infinitely many solutions. These conditions concern ranks and linear independence. Moreover, we apply a certain kind of vectorization and matrix partitioning to transform the Sylvester equation into an equivalent linear system with respect to the conventional matrix product. Our study includes the Lyapunov equation and the equation A ⋉ X = C as special cases.
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Metric geometric means with arbitrary weights of positive definite matrices involving semi-tensor products
(2023-01-01)
Ploymukda, Arnon
;
Chansangiam, Pattrawut
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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Weighted spectral geometric means and matrix equations of positive definite matrices involving semi-tensor products
(2024-01-01)
Ploymukda, Arnon
;
Tansri, Kanjanaporn
;
Chansangiam, Pattrawut
We characterized weighted spectral geometric means (SGM) of positive definite matrices in terms of certain matrix equations involving metric geometric means (MGM) ♯ and semi-tensor products ⋉. Indeed, for each real number t and two positive definite matrices A and B of arbitrary sizes, the t-weighted SGM A ⬦<inf>t</inf> B of A and B is a unique positive solution X of the equation A−1 ♯ X = (A−1 ♯ B)t. We then established fundamental properties of the weighted SGMs based on MGMs. In addition, (A ♢<inf>1/2</inf> B)2 is positively similar to A ⋉ B and, thus, they have the same spectrum. Furthermore, we showed that certain equations concerning weighted SGMs and MGMs of positive definite matrices have a unique solution in terms of weighted SGMs. Our results included the classical weighted SGMs of matrices as a special case.

Chansangiam, Pattrawut

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