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SUMMARYAn n × n real matrix A is called a bisymmetric matrix if A = A T and A = SnASn, where Sn is an n × n reverse unit matrix. This paper is mainly concerned with solving the following two problems:Problem I Given n × m real matrices X and B, and an r × r real symmetric matrix A 0 , ÿnd an n × n bisymmetric matrix A such thatwhere A([1 : r]) is a r × r leading principal submatrix of the matrix A. Problem IIGiven an n × n real matrix A * , ÿnd an n × n matrix in S E such thatwhere · is Frobenius norm, and S E is the solution set of Problem I.The necessary and su cient conditions for the existence of and the expressions for the general solutions of Problem I are given. The explicit solution, a numerical algorithm and a numerical example to Problem II are provided.

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