Speaker
Description
We study the problem of constructing solutions [1, 2] $X \in \mathbb{R}^{\alpha \times \beta}$ of a linear matrix equation
$LX = F, F \in \mathbb{R}^{\gamma \times \delta},$ (1)
where $L : \mathbb{R}^{\alpha \times \beta} \rightarrow \mathbb{R}^{\gamma \times \delta}$ is a linear bounded matrix functional. A partial case of the matrix algebraic equation (1) is the well-known Sylvester and Lyapunov equations [1, 3, 4]. In the article [5], the definition of several partial cases was introduced for a matrix algebraic equation (1) with a $\mathcal{L}$-structure
$\mathcal{L}X = F,$ (2)
which determines the linear relationship between the elements of the solution of the matrix algebraic equation. In particular, the $\mathcal{L}$-structure defines symmetric, skew-symmetric, diagonal matrices, as well as quaternions. Thus, we obtain the problem of finding solutions of a linear matrix equation (1) with a $\mathcal{L}$-structure defined by a linear bounded matrix functional
$\mathcal{L}: \mathbb{R}^{\alpha \times \beta} \rightarrow \mathbb{R}^{\lambda \times \mu} ;$
here $\mathcal{F} \in \mathbb{R}^{\lambda \times \mu}$ is a known matrix. In partial case, the $\mathcal{L}$-structure defines magic squares [6], Hilbert, Hankel and Toeplitz matrices [7], Hermitian [8], symmetric and skew-symmetric matrices, as well as quaternions and biquaternions [9, 10].
References
[1] Benner P., Bollhofer M., Kressner D., Mehl C., Stykel T. Numerical Algebra, Matrix Theory, Differential-Algebraic Equations and Co Numerical Algebra, Matrix Theory, Differential-Algebraic Equations and Control Theory. — 2015 .— Springer International Publishing. — 608 p.
[2] Chuiko S.M. The solution of the linear matrix equations // Visnyk of
V.N. Karazin Kharkiv National University. Ser. mathematics, applied mathematics and mechanics. — 2015. — 29. — P. 27 — 33.
[3] Boichuk A.A., Krivosheya S.A. Criterion of the solvability of matrix equations of the Lyapunov type // Ukrainian Mathematical Journal. – 1998.– 50 (8). –pp. 1162 – 1169.
[4] Boichuk A.A., Krivosheya S.A. A Critical Periodic Boundary Value Problem for a Matrix Riccati Equation // Differential Equations. — 2001.— 37 (4). —pp. 464 — 471.
[5] Magnus J.R. L-structured matrices and linear matrix equations //Linear and multilinear algebra. — 1983. — 14. — P. 67 — 88.
[6] Pinn K., Wieczerkowski C. Number of magic squares from parallel temperingMonte Carlo // arxiv.org, April 9, 1998.
[7] Fiedler M. Hankel and Loewner Matrices // Linear algebra and its application.— 1984. — 58. — P. 75 — 95.
[8] Khatri C.G., Mitra K. Hermitian and nonnegative definite solutions of linear matrix equations // SIAM Journal on Applied Mathematics. — 1976. — 31(4). — P. 579 — 585.
[9] Ahmad S.S., Bhadala N. L-structure least squares solutions of reduced biquaternion matrix equations with applications //ArXiv.2311.06461 DOI:10.48550.
[10] Long-Sheng Liu, Shuo Zhang A coupled quaternion matrix equations with applications // Journal of Applied Mathematics and Computing. — 2023. — 69.— P. 4069 — 4089.
