Inverse maximal eigenvalues problems for Leslie and doubly Leslie matrices

In this paper, we deal with Leslie and doubly Leslie matrices of order n. In particular, with the companion and doubly companion matrices. We study three inverse eigenvalues problems which consist of constructing these matrices from the maximal eigenvalues of its all leading principal submatrices. For Leslie and doubly companion matrices, an eigenvector associated with the maximal eigenvalue of the matrix is additionally considered, and for the doubly Leslie matrix also an eigenvector associated with the maximal eigenvalue of leading principal submatrix of order n−1 is required. We give necessary and sufficient conditions for the existence of a Leslie matrix and a companion matrix, and sufficient conditions for the existence of a doubly Leslie matrix and a doubly companion matrix. Our results are constructive and generate an algorithmic procedure to construct these special kinds of matrices.