@article{Zeng2023, 
author = {Min Zeng and Yongxin Yuan},
title = {On the solutions of the dual matrix equation        A    ⊤    X  A  =  B},
year = {2023},
journal = {Mathematical Modelling and Control},
volume = {3},
number = {3},
pages = {210-217},
keywords = {Kronecker product, dual matrix equation, optimal approximation, linear manifold},
url = {https://www.sciopen.com/article/10.3934/mmc.2023018},
doi = {10.3934/mmc.2023018},
abstract = {Let              D              m      ×      n        =  {  A  =      A          1        +  ε      A          2            |        A          1        ,      A          2        ∈            R              m      ×      n        } be the set of all    m  ×  n real dual matrices. In this paper, the following problems are considered.  Problem I: Given dual matrices    A  =      A          1        +  ε      A          2        ∈            D              m      ×      n       and    B  =      B          1        +  ε      B          2        ∈            D              n      ×      n      , find    X  ∈  S such that the dual matrix equation        A    ⊤    X  A  =  B is satisfied, where    S  =  {  X  ∈            D              m      ×      m            |    C  X  =  D  ,  C  ,  D  ∈            D              p      ×      m        }.  Problem II: Given dual matrices    A  =      A          1        +  ε      A          2        ∈            D              m      ×      n        ,  B  =      B          1        +  ε      B          2        ∈            D              n      ×      n       and              X      ~        =                    X        ~                    1        +  ε                    X        ~                    2        ∈            D              m      ×      m      , with        B          i        =      B          i        ⊤    ,  i  =  1  ,  2, find              X      ^        ∈  T such that    ‖            X      ^        −            X      ~            ‖                  D              =            min              X      ∈      T        ⁡  ‖  X  −            X      ~            ‖                  D              =            min              X      ∈      T        ⁡      ‖          X              1              −                            X          ~                            1                    ‖              2              +    ‖          X              2              −                            X          ~                            2                    ‖              2            , where    T  =  {  X  =      X          1        +  ε      X          2        ∈            D              m      ×      m            |        A    ⊤    X  A  =  B      s. t.        X          i        =      X          i        ⊤    ,  i  =  1  ,  2  }. We derive the solvability conditions and the representation of the general solution of Problem I using the Moore-Penrose inverse. Also, we deduce the solvability conditions and the explicit formula of    T and the unique approximation solution              X      ^       of Problem II by applying the Moore-Penrose inverse and Kronecker product of matrices. Finally, we give a numerical example to show the correctness of our result.}
}