(Hessenberg) eigenvalue-eigenmatrix relations

Abstract

Explicit relations between eigenvalues, eigenmatrix entries and matrix elements are derived. First, a general, theoretical result based on the Taylor expansion of the adjugate of zI - A on the one hand and explicit knowledge of the Jordan decomposition on the other hand is proven. This result forms the basis for several, more practical and enlightening results tailored to non-derogatory, diagonalizable and normal matrices, respectively. Finally, inherent properties of (upper) Hessenberg, resp. tridiagonal matrix structure are utilized to construct computable relations between eigenvalues, eigenvector components, eigenvalues of principal submatrices and products of lower diagonal elements.