[{"data":1,"prerenderedAt":-1},["ShallowReactive",2],{"doc-detail-83173-en":3,"doc-seo-83173-105":29,"detail-sidebar-cat-0-en-105":91},{"code":4,"msg":5,"data":6},0,"success",{"doc_id":7,"user_id":8,"nickname":9,"user_avatar":10,"doc_module":4,"category_id":11,"category_name":12,"doc_title":13,"doc_description":14,"doc_content":15,"file_id":16,"file_url":17,"file_type":18,"file_size":19,"view_count":20,"is_deleted":4,"is_public":20,"is_downloadable":20,"audit_status":20,"page_count":21,"language":22,"language_code":23,"site_id":24,"html_lang":23,"table_of_contents":25,"faqs":26,"seo_title":13,"seo_description":14,"update_tm":27,"read_time":28},83173,2336464648746,"Skyler","https://ap-avatar.wpscdn.com/davatar_276721f389ce27ea32af1340a28f341c",8,"Research & Report","Solution Analysis of Tensor Equation A ⋉ X ⋉ B = C via Semi Tensor Product with t-product","The paper analyzes the tensor equation A ⋉ X ⋉ B = C using a semi tensor product framework combined with the t-product. For an unknown vector X, it derives a necessary and sufficient solvability condition, giving a precise equivalence criterion for existence of solutions. For matrix-valued and higher-order tensor-valued X, solvability is characterized by corresponding compatibility requirements. The work also identifies the explicit Toeplitz and Circulant structure of C and validates the results through illustrative examples.","arXiv :2607 .07006v1 [math .NA] 8 Jul 2026  \nSolution Analysis of Tensor Equation A ⋉ X ⋉ B = C via Semi Tensor Product with t-product  \nBhawna Garg  Ranjan Kumar Das †  \nAbstract  \nThis paper focuses on the analysis of the tensor equation A ⋉ X ⋉ B = C, formulated via the semi tensor product with t-product. For the unknown vector X, we establish a necessary and sufficient condition that provides an equivalence criterion for the existence of solutions. For matrix valued and higher-order tensor valued unknown X, solvability is determined by corresponding compatibility requirements. Moreover, the explicit structure (Toeplitz and Circulant) of C is characterized. The derived results are supported by several illustrative examples.  \nKeywords: Tensor Equation, Semi Tensor Product, t-product, Compatibility Conditions, Toeplitz and Circulant Tensor.  \nMathematics Subject Classification: 15A69, 15A60 .  \n1 Introduction  \nTensors provide a powerful and natural framework for representing and analyzing multiway data arising in modern scientific and engineering applications, see [1, 2, 10, 20, 24, 26, 27] and the references therein. Unlike matrices, tensors are capable of preserving intrinsic multidimensional structures and correlations, which renders them essential in applications such as signal processing, image and video analysis, machine learning, control systems, and networked dynamical systems [4, 14, 16, 25, 30] . As the dimensionality and complexity of data continue to increase, tensor-based modeling has become increasingly important.  \nThe algebraic manipulation of tensors relies heavily on the choice of tensor products. In last decades, various tensor products have been introduced in the literature, including the Kronecker product [3], k-mode product [28], Einstein product [11], Tucker product [20], and the t-product [19] . Among these, the t-product gain significant attention due to its strong algebraic properties, ability to preserve matrix-like structures, and computational efficiency through block circulant  \n∗ Department of Mathematics, NIT Raipur, Raipur-492010, India([bhawna1601@gmail.com](bhawna1601@gmail.com)).  \n†Corresponding author, Department of Mathematics, NIT Raipur, Raipur-492010, India([rkdas.maths@nitrr.ac.in](rkdas.maths@nitrr.ac.in)).  \nrepresentations and fast Fourier transforms. However, most existing tensor products, includingthe standard t-product, impose strict dimensional compatibility requirements, which limit their applicability in practical problems involving tensor dimension.  \nTo overcome dimensional incompatibility, Cheng et al. [8] introduced the semi-tensor product (STP), which generalizes the conventional matrix product by embedding matrices into higher-dimensional spaces via the Kronecker product. The STP coincides with the standard matrix multiplication when dimensions are compatible, while remaining well defined for arbitrary dimensions. Due to its flexibility, the STP has become an important algebraic tool in control theory, Boolean networks, game theory, and nonlinear dynamical systems, and has been successfully extended from matrices to higher-order tensors [12, 23, 22] .  \nMatrix equations of the form AX = C play a fundamental role in systems theory and numerical analysis [12, 29] . Under the STP framework, this classical equation has been extended to cases with incompatible dimensions, and further generalized to tensor equations, allowing the unknown variable X to be a vector, matrix, or tensor form. These developments have significantly broadened the scope of solvable linear systems in multilinear algebra [12] .  \nBeyond the equation AX = C, more general matrix equation of the form AXB = C have also been extensively studied by employing the STP approach. Several studies have investigated existence conditions and computational approaches for the matrix equation A ⋉ X ⋉ B = C , where ⋉ is referred as semi-tensor product and the matrices A, B , C are given and the matrix X is to be deter","cbCaitCGXukPZIs7","https://ap.wps.com/l/cbCaitCGXukPZIs7","pdf",465827,1,23,"English","en",105,"# Introduction\n# Preliminaries","[{\"question\":\"What equation does the paper study and what is the goal of the analysis?\",\"answer\":\"The paper studies the tensor equation A ⋉ X ⋉ B = C. It focuses on determining when solutions exist and how those solutions can be characterized under the semi tensor product with t-product framework.\"},{\"question\":\"What solvability condition is obtained for the unknown vector X?\",\"answer\":\"For vector-valued X, the paper establishes a necessary and sufficient condition that serves as an equivalence criterion for the existence of solutions.\"},{\"question\":\"How are solvability and the structure of C characterized for more general X?\",\"answer\":\"For matrix-valued and higher-order tensor-valued X, solvability is determined by corresponding compatibility requirements. In addition, the explicit Toeplitz and Circulant structure of C is characterized, and supported by examples.\"}]",1784185741,58,{"code":4,"msg":30,"data":31},"ok",{"site_id":24,"language":23,"slug":32,"title":13,"keywords":33,"description":14,"schema_data":34,"social_meta":86,"head_meta":88,"extra_data":90,"updated_unix":27},"solution-analysis-of-tensor-equation-a-x-b-c-via-semi-tensor-product-with-t-product","",{"@graph":35,"@context":85},[36,53,68],{"@type":37,"itemListElement":38},"BreadcrumbList",[39,43,47,50],{"item":40,"name":41,"@type":42,"position":20},"https://docshare.wps.com","Home","ListItem",{"item":44,"name":45,"@type":42,"position":46},"https://docshare.wps.com/document/","Document",2,{"item":48,"name":12,"@type":42,"position":49},"https://docshare.wps.com/document/research-report/",3,{"item":51,"name":13,"@type":42,"position":52},"https://docshare.wps.com/document/solution-analysis-of-tensor-equation-a-x-b-c-via-semi-tensor-product-with-t-product/83173/",4,{"url":51,"name":13,"@type":54,"author":55,"headline":13,"publisher":57,"fileFormat":60,"inLanguage":23,"description":14,"dateModified":61,"datePublished":62,"encodingFormat":60,"isAccessibleForFree":63,"interactionStatistic":64},"DigitalDocument",{"name":9,"@type":56},"Person",{"url":40,"name":58,"@type":59},"DocShare","Organization","application/pdf","2026-07-17","2026-07-16",true,{"@type":65,"interactionType":66,"userInteractionCount":20},"InteractionCounter",{"@type":67},"ViewAction",{"@type":69,"mainEntity":70},"FAQPage",[71,77,81],{"name":72,"@type":73,"acceptedAnswer":74},"What equation does the paper study and what is the goal of the analysis?","Question",{"text":75,"@type":76},"The paper studies the tensor equation A ⋉ X ⋉ B = C. It focuses on determining when solutions exist and how those solutions can be characterized under the semi tensor product with t-product framework.","Answer",{"name":78,"@type":73,"acceptedAnswer":79},"What solvability condition is obtained for the unknown vector X?",{"text":80,"@type":76},"For vector-valued X, the paper establishes a necessary and sufficient condition that serves as an equivalence criterion for the existence of solutions.",{"name":82,"@type":73,"acceptedAnswer":83},"How are solvability and the structure of C characterized for more general X?",{"text":84,"@type":76},"For matrix-valued and higher-order tensor-valued X, solvability is determined by corresponding compatibility requirements. In addition, the explicit Toeplitz and Circulant structure of C is characterized, and supported by examples.","https://schema.org",{"og:url":51,"og:type":87,"og:title":13,"og:site_name":58,"og:description":14},"article",{"robots":89,"canonical":51},"index,follow",{"doc_id":7,"site_id":24},{"code":4,"msg":5,"data":92},[93,97,101,105,110,115,120,123,128,131,135],{"id":20,"doc_module":4,"doc_module_name":45,"category_name":94,"show_sort_weight":95,"slug":96},"Story & Novel",90,"story-novel",{"id":46,"doc_module":4,"doc_module_name":45,"category_name":98,"show_sort_weight":99,"slug":100},"Literature",80,"literature",{"id":52,"doc_module":4,"doc_module_name":45,"category_name":102,"show_sort_weight":103,"slug":104},"Exam",70,"exam",{"id":106,"doc_module":4,"doc_module_name":45,"category_name":107,"show_sort_weight":108,"slug":109},5,"Comic",60,"comic",{"id":111,"doc_module":4,"doc_module_name":45,"category_name":112,"show_sort_weight":113,"slug":114},6,"Technology",50,"technology",{"id":116,"doc_module":4,"doc_module_name":45,"category_name":117,"show_sort_weight":118,"slug":119},7,"Healthcare",40,"healthcare",{"id":11,"doc_module":4,"doc_module_name":45,"category_name":12,"show_sort_weight":121,"slug":122},30,"research-report",{"id":124,"doc_module":4,"doc_module_name":45,"category_name":125,"show_sort_weight":126,"slug":127},9,"Religion & Spirituality",20,"religion-spirituality",{"id":126,"doc_module":4,"doc_module_name":45,"category_name":129,"show_sort_weight":126,"slug":130},"World Cup","world-cup",{"id":132,"doc_module":4,"doc_module_name":45,"category_name":133,"show_sort_weight":132,"slug":134},10,"Lifestyle","lifestyle",{"id":136,"doc_module":4,"doc_module_name":45,"category_name":137,"show_sort_weight":106,"slug":138},19,"General","general"]