An analysis of the HR algorithm for computing the eigenvalues of a matrix

AbstractThe HR algorithm, a method of computing the eigenvalues of a matrix, is presented. It is based on the fact that almost every complex square matrix A can be decomposed into a product A = HR of a so-called pseudo-Hermitian matrix H and an upper triangular matrix R. This algorithm is easily seen to be a generalization of the well-known QR algorithm. It is shown how it is related to the power method and inverse iteration, and for special matrices the connection between the LR and HR algorithms is indicated

Unital Quantum Channels - Convex Structure and Revivals of Birkhoff's Theorem

The set of doubly-stochastic quantum channels and its subset of mixtures of unitaries are investigated. We provide a detailed analysis of their structure together with computable criteria for the separation of the two sets. When applied to O(d)-covariant channels this leads to a complete characterization and reveals a remarkable feature: instances of channels which are not in the convex hull of unitaries can return to it when either taking finitely many copies of them or supplementing with a completely depolarizing channel. In these scenarios this implies that a channel whose noise initially resists any environment-assisted attempt of correction can become perfectly correctable.Comment: 31 page

Zero dynamics and stabilization for linear DAEs

We study linear differential-algebraic multi-input multi-output systems which are not necessarily regular and investigate the asymptotic stability of the zero dynamics and stabilizability. To this end, the concepts of autonomous zero dynamics, transmission zeros, right-invertibility, stabilizability in the behavioral sense and detectability in the behavioral sense are introduced and algebraic characterizations are derived. It is then proved, for the class of right-invertible systems with autonomous zero dynamics, that asymptotic stability of the zero dynamics is equivalent to three conditions: stabilizability in the behavioral sense, detectability in the behavioral sense, and the condition that all transmission zeros of the system are in the open left complex half-plane. Furthermore, for the same class, it is shown that we can achieve, by a compatible control in the behavioral sense, that the Lyapunov exponent of the interconnected system equals the Lyapunov exponent of the zero dynamics

Error bounds in the isometric Arnoldi process

Error bounds for the eigenvalues computed in the isometric Arnoldi method are derived. The Arnoldi method applied to a unitary matrix U successively computes a sequence of unitary upper Hessenberg matrices Hk ; k = 1; 2; : : :. The eigenvalues of the Hk 's are increasingly better approximations to eigenvalues of U . An upper bound for the distance of the spectrum of Hk from the spectrum of U , and an upper bound for the distance between each individual eigenvalue of Hk and one of U are given. Between two eigenvalues of Hk on the unit circle, there is guaranteed to lie an eigenvalue of U . The results are applied to a problem in signal processing. Key words. unitary eigenvalue problem, Arnoldi process, error bounds, signal processing AMS(MOS) subject classifications. 65F15, 15A18 1 Introduction A number of signal processing problems can be seen to require the numerical solution of unitary eigenvalue problems. The task of estimating dominant harmonics of a time series for instances ar..

Regularization of descriptor systems by output feedback

Conditions are given under which a descriptor, or generalized state-space system can be regularized by output feedback. It is shown that under these conditions, proportional and derivative output feedback controls can be constructed such that the closed-loop system is regular and has index at most one. This property ensures the solvability of the resulting system of dynamic-algebraic equations. A reduced form is given that allows the system properties as well as the feedback to be determined. The construction procedures used to establish the theory are based only on orthogonal matrix decompositions and can therefore be implemented in a numerically stable way

On The Perturbation Theory For Unitary Eigenvalue Problems

. Some aspects of the perturbation theory for eigenvalues of unitary matrices are considered. Making use of the close relation between unitary and Hermitian eigenvalue problems a Courant-Fischer-type theorem for unitary matrices is derived and an inclusion theorem analogue to the Kahan theorem for Hermitian matrices is presented. Implications for the special case of unitary Hessenberg matrices are discussed. Key words. unitary eigenvalue problem, perturbation theory AMS(MOS) subject classifications. 15A18, 65F99 1. Introduction. New numerical methods to compute eigenvalues of unitary matrices have been developed during the last ten years. Unitary QR-type methods [19, 9], a divide-and-conquer method [20, 21], a bisection method [10], and some special methods for the real orthogonal eigenvalue problem [1, 2] have been presented. Interest in this task arose from problems in signal processing [11, 29, 33], in Gaussian quadrature on the unit circle [18], and in trigonometric approximation..