## Least-squares solutions of matrix inverse problem for bi-symmetric matrices with a submatrix constraint.(English)Zbl 1199.65141

Summary: An $$n\times n$$ real matrix $$A = (a_{ij})_{n\times n}$$ is called bi-symmetric matrix if $$A$$ is both symmetric and per-symmetric, that is, $$a_{ij} = a_{ji}$$ and $$a_{ij} = a_{n+1-j,n+1-i}$$ $$(i,j = 1, 2,\dots , n)$$. This paper is mainly concerned with finding the least-squares bi-symmetric solutions of matrix inverse problem $$AX = B$$ with a submatrix constraint, where $$X$$ and $$B$$ are given matrices of suitable sizes. Moreover, in the corresponding solution set, the analytical expression of the optimal approximation solution to a given matrix $$A^*$$ is derived. A direct method for finding the optimal approximation solution is described in detail, and three numerical examples are provided to show the validity of our algorithm.

### MSC:

 65F30 Other matrix algorithms (MSC2010) 15A24 Matrix equations and identities 65F20 Numerical solutions to overdetermined systems, pseudoinverses
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