@Article{NM-16-215, author = { Y. Y. Zhang, Y. Lei and A. P. Liao}, title = {Least-Squares Solutions of the Matrix Equation A^TXA=B Over Bisymmetric Matrices and its Optimal Approximation}, journal = {Numerical Mathematics, a Journal of Chinese Universities}, year = {2007}, volume = {16}, number = {3}, pages = {215--225}, abstract = { A real $n\times n$ symmetric matrix $X=(x_{ij})_{n \times n}$ is called a bisymmetric matrix if $x_{ij}=x_{n+1-j, n+1-i}$. Based on the projection theorem, the canonical correlation decomposition and the generalized singular value decomposition, a method useful for finding the least-squares solutions of the matrix equation $A^{T}XA=B$ over bisymmetric matrices is proposed. The expression of the least-squares solutions is given. Moreover, in the corresponding solution set, the optimal approximate solution to a given matrix is also derived. A numerical algorithm for finding the optimal approximate solution is also described.}, issn = {}, doi = {https://doi.org/}, url = {http://global-sci.org/intro/article_detail/nm/8055.html} }