Inverse eigenvalue problems of tridiagonal symmetric matrices and tridiagonal bisymmetric matrices

The problem of generating a matrix A with specified eigenpairs, where A is a tridiagonal symmetric matrix, is presented. A general expression of such a matrix is provided, and the set of such matrices is denoted by SE. Moreover, the corresponding least-squares problem under spectral constraint is considered when the set SE is empty, and the corresponding solution set is denoted by SL. The best approximation problem associated with SE(SL) is discussed, that is: to find the nearest matrix in SE(SL) to a given matrix. The existence and uniqueness of the best approximation are proved and the expression of this nearest matrix is provided. At the same time, we also discuss similar problems when A is a tridiagonal bisymmetric matrix.