Wavelet and Linear Algebra

Abstract This paper explores the Lyapunov, quadratic matrix, and matrix square root equations under normality assumptions on the system matrix, utilizing the Moore-Penrose inverse ($A^\dagger$) for solvability in singular cases. We provide a detailed historical overview and extend our spectral decomposition method to achieve unique solutions for these equations, building on prior work for invertible matrices. The proposed approach handles singular normal matrices via orthogonality conditions on the kernel, and numerical examples confirm its accuracy and efficiency compared to conventional methods.

Abstract This paper addresses the solvability of the tensor system A⋉X = B, X ⋉C = D using the
semi-tensor product framework. Initially, we examine the associated tensor-vector system
under the semi-tensor product operation, deriving necessary and sufficient conditions for
the existence of solutions and propose explicit solution methods. Furthermore, we introduce suitable tensor partitioning techniques that enable the reduction of the tensor-vector
system to a linear system formulated via the t-product. This approach facilitates a comprehensive study of the solvability of the tensor system when the unknown tensor X is a
vector, matrix or a higher-order tensor under the semi-tensor product. We apply the STP
framework to model and control Boolean gene regulatory networks with pinning control
inputs. Several illustrative examples are provided to demonstrate the applicability and effectiveness of the proposed approaches. Our study includes the tensor equation A⋉X = B
and the matrix system A ⋉ X = B, X ⋉ C = D as special cases.

Abstract In this paper, by employing the concept of operator ranges and their equality, and utilizing the invertibility of certain operators, we establish the equality of ranges of related operators in Hilbert
\‎(C^{*}\)-modules. The concept of operator ranges, combined with the invertibility of these operators, provides a tool for identifying and proving range equalities of related operators.‎

Abstract ‎In this paper‎, ‎we investigate the automatic continuity of linear mappings and homomorphisms on certain class of fundamental F-algebras and more generally on topological algebras‎. ‎Some examples illustrating these results are presented as well.‎In this paper‎, ‎we investigate the automatic continuity of linear mappings and homomorphisms on certain class of fundamental F-algebras and more generally on topological algebras‎. Some examples illustrating these results are presented as wel