We survey the quadratic eigenvalue problem, treating its many applications, its mathematical properties, and a variety of numerical solution techniques. Emphasis is given to exploiting both the structure of the matrices in the problem (dense, sparse, real, complex, Hermitian, skew-Hermitian) and the spectral properties of the problem. We classify numerical methods and catalogue available software.

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The Quadratic Eigenvalue Problem

Quaternions and matrices of quaternions

This balanced and comprehensive study presents the theory, methods and applications of matrix analysis in a new theoretical framework, allowing readers to understand second-order and higher-order matrix analysis in a completely new light Alongside the core subjects in matrix analysis, such as singular value analysis, the solution of matrix equations and eigenanalysis, the author introduces new applications and perspectives that are unique to this book The very topical subjects of gradient analysis and optimization play a central role here Also included are subspace analysis, projection analysis and tensor analysis, subjects which are often neglected in other books Having provided a solid foundation to the subject, the author goes on to place particular emphasis on the many applications matrix analysis has in science and engineering, making this book suitable for scientists, engineers and graduate students alike

Matrix Analysis and Applications

http://www.gipsa-lab.grenoble-inp.fr/~jerome.mars/2004_Signal_Processing_Quater_NLB.pdf

Singular value decomposition of quaternion matrices: a new tool for vector-sensor signal processing

We study large-scale, continuous-time linear time-invariant control systems with a sparse or structured state matrix and a relatively small number of inputs and outputs. The main contributions of this paper are numerical algorithms for the solution of large algebraic Lyapunov and Riccati equations and linearquadratic optimal control problems, which arise from such systems. First, we review an alternating direction implicit iteration-based method to compute approximate low-rank Cholesky factors of the solution matrix of large-scale Lyapunov equations, and we propose a refined version of this algorithm. Second, a combination of this method with a variant of Newton's method (in this context also called Kleinman iteration) results in an algorithm for the solution of large-scale Riccati equations. Third, we describe an implicit version of this algorithm for the solution of linear-quadratic optimal control problems, which computes the feedback directly without solving the underlying algebraic Riccati equation explicitly. Our algorithms are efficient with respect to both memory and computation. In particular, they can be applied to problems of very large scale, where square, dense matrices of the system order cannot be stored in the computer memory. We study the performance of our algorithms in numerical experiments.

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Numerical solution of large‐scale Lyapunov equations, Riccati equations, and linear‐quadratic optimal control problems

A method is presented to solve the real algebraic Riccati equation - XNX + XA + A^{T}X + K = 0 , where K = K^{T} and N = N^{T} . The solution for the corresponding eigenvalue problem Mx = \lambda x , where M is a Hamiltonian matrix, is computed by an algorithm similar to the QR algorithm. Special symplectic matrices are used for the transformation of M such that the Hamiltonian form is preserved during the computations.

A symplectic QR like algorithm for the solution of the real algebraic Riccati equation

Matrix factorizations for symplectic QR-like methods

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

Schur parameter pencils for the solution of the unitary eigenproblem

This paper extends the Francis QR algorithm to quaternion and antiquaternion matrices. It calculates a quaternion version of the Schur decomposition using quaternion unitary similarity transformations. Following a finite step reduction to a Hessenberg-like condensed form, a sequence of implicit QR steps reduces the matrix to triangular form. Eigenvalues may be read off the diagonal. Eigenvectors may be obtained from simple back substitutions. For serial computation, the algorithm uses only half the work and storage of the unstructured Francis QR iteration. By preserving quaternion structure, the algorithm calculates the eigenvalues of a nearby quaternion matrix despite rounding errors.

Angelika Bunse-Gerstner

Papers

A symplectic QR like algorithm for the solution of the real algebraic Riccati equation

Matrix factorizations for symplectic QR-like methods

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

Schur parameter pencils for the solution of the unitary eigenproblem

A quaternion QR algorithm