On the generalized bi (skew-) symmetric solutions of a linear matrix equation and its procrust problems

In this paper, the solvability conditions and the explicit expressions of the generalized bisymmetric and bi-skew-symmetric solutions of the matrix equation AX=BAX=B are respectively established by applying two methods. Then the maximal and minimal ranks of the solutions are derived. If the solvability conditions are not satisfied, the generalized bisymmetric and bi-skew-symmetric least squares solutions of the matrix equation are considered, and the generalized bisymmetric and bi-skew-symmetric least squares solutions with the minimum norm are also obtained. In addition, two algorithms are provided to compute the generalized bi (skew-) symmetric least squares solution, and some examples are given to illustrate that the algorithms are feasible.