Hessenberg eigenvalue–eigenmatrix relations

Explicit relations between eigenvalues, eigenmatrix entries and matrix elements of unreduced Hessenberg matrices are derived. The main result is based on the Taylor expansion of the adjugate of zI-H on the one hand and inherent properties of Hessenberg matrix structure on the other hand. This result is utilized to construct computable relations between eigenvalues, eigenvector components, eigenvalues of principal submatrices and products of lower diagonal elements, generalizing similar identities for Jacobi matrices.