On a Special Inverse Singular Value Problem for Certain Classes of Hessenberg Matrices

This paper addresses an inverse singular value problem for three classes of upper Hessenberg matrices: Leslie-type, companion, and circular bidiagonal matrices. The problem is to construct such matrices from the prescribed extremal singular values of certain submatrices. A connection with known results on inverse extremal eigenvalue problems for bordered diagonal and periodic Jacobi matrices is established. For all matrices, conditions for the existence and uniqueness of solutions are provided, except for the signs of their entries. In the case of Leslie-type and companion matrices, these conditions are necessary and sufficient. For circular bidiagonal matrices, these conditions are only sufficient, and two admissible auxiliary parameters are required. Furthermore, numerical examples illustrate the proposed constructions.