Solution Analysis of Tensor Equation $\mathcal{A} \ltimes \mathcal{X} \ltimes \mathcal{B}= \mathcal{C}$ via Semi Tensor Product with t-product
Year: 2026
Type: preprint
Abstract: Tensor equations involving both left and right tensor operators arise naturally in applications where multidimensional data are coupled through transformations acting from both sides. Motivated by the need to analyze such equations, this manuscript investigates a detailed solution analysis of $\mathcal{A}\ltimes \mathcal{X}\ltimes \mathcal{B} = \mathcal{C}$ via the semi-tensor product with the $t$-product. The manuscript provides a more general framework that includes, as a special case, the tensor equation $\mathcal{A}\ltimes\mathcal{X}=\mathcal{B}$ considered in \cite{J.FathitensorequationAX=BunderSTP}. Necessary and sufficient conditions are derived for the existence of vector and matrix valued solutions. Explicit compatibility conditions are established, and constructive Moore-Penrose-inverse-based algorithms are provided for the vector and matrix valued cases. The computational complexity and execution times are compared for Discrete Fourier Transform(DFT) and Fast Fourier Transform(FFT) based \(t\)-product implementations. The practical relevance of the proposed framework is further demonstrated through a color image deblurring application. Examples throughout illustrate the theoretical results.
Source: arXiv (Cornell University)
Authors: Garg, Bhawna, Das, Ranjan Kumar
Institution: National Institute of Technology Raipur
Language: English
Subfield: Computational Mathematics
Field: Mathematics
Domain: Physical Sciences
Open Access status: green
OpenAlex ID: W7167808309