Condensed Forms for Skew-Hamiltonian/Hamiltonian Pencils

Condensed Forms for Skew-Hamiltonian/Hamiltonian Pencils

Abstract In this paper we consider real or complex skew-Hamiltonian/Hamiltonian pencils λS −H, i.e., pencils where S is a skew-Hamiltonian and H is a Hamiltonian matrix. These pencils occur for example in the theory of continuous time, linear quadratic optimal control problems. We reduce these pencils to canonical and Schur-type forms under structure-preserving transformations, i.e., J-congruen...

A Structured Staircase Algorithm for Skew-symmetric/symmetric Pencils

A STRUCTURED STAIRCASE ALGORITHM FOR SKEW-SYMMETRIC/SYMMETRIC PENCILS RALPH BYERS , VOLKER MEHRMANN , AND HONGGUO XU Abstract. We present structure preserving algorithms for the numerical computation of structured staircase forms of skew-symmetric/symmetric matrix pencils along with the Kronecker indices of the associated skew-symmetric/symmetric Kronecker-like canonical form. These methods all...

Reduction and Normal Forms of Matrix Pencils

Matrix pencils, or pairs of matrices, may be used in a variety of applications. In particular, a pair of matrices (E,A) may be interpreted as the differential equation Ex′ + Ax = 0. Such an equation is invariant by changes of variables, or linear combination of the equations. This change of variables or equations is associated to a group action. The invariants corresponding to this group action...

Numerical Computation of Deflating Subspaces of Skew-Hamiltonian/Hamiltonian Pencils

We discuss the numerical solution of structured generalized eigenvalue problems that arise from linear-quadratic optimal control problems, H∞ optimization, multibody systems, and many other areas of applied mathematics, physics, and chemistry. The classical approach for these problems requires computing invariant and deflating subspaces of matrices and matrix pencils with Hamiltonian and/or ske...

Apolarity of Trilinear Forms and Pencils of Bilinear Forms*