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Quaternionic conjugate-gradient method for solving the matrix equation AXB=C over generalized quaternions
(2026-10-01)
Tansri, Kanjanaporn
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Chansangiam, Pattrawut
;
Zhang, Yang
We propose a structure-exploiting conjugate-gradient (CG)–type algorithm for solving the matrix equation AXB=C over the generalized quaternions. The proposed method is developed from an idea of operating the linear map K(X)=AXB directly on the matrix space. This enables a matrix-free Krylov subspace implementation that avoids the explicit construction of the associated large-scale Kronecker matrix. By exploiting the intrinsic component-wise structure of quaternion matrices, the procedure performs all computations through matrix–matrix multiplications, leading to significant reductions in computational complexity and memory requirements. Finite-step convergence of the algorithm is established under suitable positive-definiteness assumptions. The proposed framework naturally includes real- and complex-valued matrix equations, the Hamilton quaternion case, and other quaternion algebras as special cases. Numerical experiments confirm the efficiency, robustness, and scalability of the proposed algorithm compared with existing methods.
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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
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Phoonphiphat, Thanaphon
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Chansangiam, Pattrawut
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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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The Nature and Significance of Mathematics from Contemporary Viewpoints
(2025-01-01)
Boonnam, Nathaphon
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Hama, Rattanasak
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Chansangiam, Pattrawut
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Sabau, Sorin V.
This paper explores the nature and significance of mathematics, presenting it as a systematic and logical study of patterns in nature. Mathematics can be conceptualized as a pyramid consisting of three layers. The first layer is pure mathematics, which focuses on the study of abstract objects and concepts. The second layer is applied mathematics, dedicated to the development and application of mathematical methods to address specific problems. The final layer involves the applications of mathematics, where established results from pure or applied mathematics are utilized to solve concrete, real-world problems. A deep appreciation of the importance and beauty of abstract patterns requires engaging in research within pure or applied mathematics. To undertake such research, the fundamentals of pure mathematics are essential. Advances in mathematical research often lead to the development of new theories or innovative techniques for problem-solving.
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Exact and least-squares solutions of a generalized Sylvester-transpose matrix equation over generalized quaternions
(2024-01-01)
Jaiprasert, Janthip
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Chansangiam, Pattrawut
We have considered a generalized Sylvester-transpose matrix equation AXB + CXTD = E, where A, B,C, D, and E are given rectangular matrices over a generalized quaternion skew-field, and X is an unknown matrix. We have applied certain vectorizations and real representations to transform the matrix equation into a matrix equation over the real numbers. Thus, we have investigated a solvability condition, general exact/least-squares solutions, minimal-norm solutions, and the exact/least-squares solution closest to a given matrix. The main equation included the equation AXB = E and the Sylvester-transpose equation. Our results also covered such matrix equations over the quaternions, and quaternionic linear systems.
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Weighted spectral geometric means and matrix equations of positive definite matrices involving semi-tensor products
(2024-01-01)
Ploymukda, Arnon
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Tansri, Kanjanaporn
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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.
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Gradient-descent iterative algorithm for solving exact and weighted least-squares solutions of rectangular linear systems
(2023-01-01)
Tansri, Kanjanaporn
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Chansangiam, Pattrawut
Consider a linear system Ax = b where the coefficient matrix A is rectangular and of full-column rank. We propose an iterative algorithm for solving this linear system, based on gradient-descent optimization technique, aiming to produce a sequence of well-approximate least-squares solutions. Here, we consider least-squares solutions in a full generality, that is, we measure any related error through an arbitrary vector norm induced from weighted positive definite matrices W. It turns out that when the system has a unique solution, the proposed algorithm produces approximated solutions converging to the unique solution. When the system is inconsistent, the sequence of residual norms converges to the weighted least-squares error. Our work includes the usual least-squares solution when W = I. Numerical experiments are performed to validate the capability of the algorithm. Moreover, the performance of this algorithm is better than that of recent gradient-based iterative algorithms in both iteration numbers and computational time.
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Numerical solutions of the space-time fractional diffusion equation via a gradient-descent iterative procedure
(2023-01-01)
Tansri, Kanjanaporn
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Kittisopaporn, Adisorn
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Chansangiam, Pattrawut
A one-dimensional space-time fractional diffusion equation describes anomalous diffusion on fractals in one dimension. In this paper, this equation is discretized by finite difference schemes based on the Grünwald-Letnikov approximation for Riemann-Liouville and Caputo’s fractional derivatives. It turns out that the discretized equations can be put into a compact form, i.e., a linear system with a block lower-triangular coefficient matrix. To solve the linear system, we formulate a matrix iterative algorithm based on gradient-descent technique. In particular, we work out for the space fractional diffusion equation. Theoretically, the proposed solver is always applicable with satisfactory convergence rate and error estimates. Simulations are presented numerically and graphically to illustrate the accuracy, the efficiency, and the performance of the algorithm, compared to other iterative procedures for linear systems.
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Riccati equation and metric geometric means of positive semidefinite matrices involving semi-tensor products
(2023-01-01)
Chansangiam, Pattrawut
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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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Metric geometric means with arbitrary weights of positive definite matrices involving semi-tensor products
(2023-01-01)
Ploymukda, Arnon
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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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Conjugate Gradient Algorithm for Least-Squares Solutions of a Generalized Sylvester-Transpose Matrix Equation
(2022-09-01)
Tansri, Kanjanaporn
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Chansangiam, Pattrawut
We derive a conjugate-gradient type algorithm to produce approximate least-squares (LS) solutions for an inconsistent generalized Sylvester-transpose matrix equation. The algorithm is always applicable for any given initial matrix and will arrive at an LS solution within finite steps. When the matrix equation has many LS solutions, the algorithm can search for the one with minimal Frobenius-norm. Moreover, given a matrix Y, the algorithm can find a unique LS solution closest to Y. Numerical experiments show the relevance of the algorithm for square/non-square dense/sparse matrices of medium/large sizes. The algorithm works well in both the number of iterations and the computation time, compared to the direct Kronecker linearization and well-known iterative methods.

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