Solution Analysis of Tensor Equation 𝒜 ltimes 𝒳 ltimes ℬ= 𝒞 via Semi Tensor Product with t-product
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 𝒜ltimes 𝒳ltimes ℬ = 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 𝒜ltimes𝒳=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.
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BibTeX
@misc{garg2026solution,
title = {{Solution Analysis of Tensor Equation $\mathcalA \ltimes \mathcalX \ltimes \mathcalB= \mathcalC$ via Semi Tensor Product with t-product}},
author = {Bhawna Garg and Ranjan Kumar Das},
howpublished = {arXiv},
year = {2026},
month = sep,
doi = {10.48550/arxiv.2607.07006},
eprint = {2607.07006},
archivePrefix = {arXiv},
url = {https://arxiv.org/abs/2607.07006},
}
RIS
TY - UNPB
TI - Solution Analysis of Tensor Equation 𝒜 ltimes 𝒳 ltimes ℬ= 𝒞 via Semi Tensor Product with t-product
AU - Bhawna Garg
AU - Ranjan Kumar Das
PY - 2026
DA - 2026/09/25
PB - arXiv
DO - 10.48550/arxiv.2607.07006
UR - https://arxiv.org/abs/2607.07006
AB - 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 𝒜ltimes 𝒳ltimes ℬ = 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 𝒜ltimes𝒳=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.
ER -