Solution of the matrix eigenvalue problem <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" altimg="si7.gif" overflow="scroll"><mml:mtext>VA</mml:mtext><mml:mo>+</mml:mo><mml:msup><mml:mrow><mml:mi>A</mml:mi></mml:mrow><mml:mrow><mml:mo>*</mml:mo></mml:mrow></mml:msup><mml:mi>V</mml:mi><mml:mo>=</mml:mo><mml:mi>μ</mml:mi><mml:mi>V</mml:mi></mml:math> with applications to the study of free linear dynamical systems

Kohaupt, L.

doi:10.1016/j.cam.2007.01.001

Cited by 13 publications

(2 citation statements)

References 26 publications

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“…Finally, Section 9 contains the conclusions. The non-cited references [ 4], [ 5], [ 6], [ 7], [ 8], [ 9], [10], and [11] are given because they may be useful to the reader in the context of the present paper.…”

Section: Introductionmentioning

confidence: 99%

Rayleigh-Quotient Representation of the Real Parts, Imaginary Parts, and Moduli of the Eigenvalues of General Matrices

KOHAUPT

2020

Journal of Mathematical Sciences and Modelling

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In the present paper, formulas for the Rayleigh-quotient representation of the real parts, imaginary parts, and moduli of the eigenvalues of general matrices are obtained that resemble corresponding formulas for the eigenvalues of self-adjoint matrices. These formulas are of interest in Linear Algebra and in the theory of linear dynamical systems. The key point is that a weighted scalar product is used that is defined by means of a special positive definite matrix. As applications, one obtains convexity properties of newly-defined numerical ranges of a matrix. A numerical example underpins the theoretical findings.

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Section: Introductionmentioning

confidence: 99%

Rayleigh-Quotient Representation of the Real Parts, Imaginary Parts, and Moduli of the Eigenvalues of General Matrices

KOHAUPT

2020

Journal of Mathematical Sciences and Modelling

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“…A periodic system (1) can be transformed via the Floquet-Lyapunov transformation into a set of linear ODEs with constant coefficientsẏ = Ry with y(0) = x 0 . Since Z(t) is bounded, the original system (1) can be bounded by the spectral abscissa c 1 e −ν(−R)t ≤ x(t) ∞ ≤ C 1 e ν(R)t and by [5] for a diagonalizable R where λ i ∈ Λ(R) are the eigenvalues, u i the associated left eigenvectors and…”

Section: Rigorous Boundsmentioning

confidence: 99%

Bounds on the Solution of Linear Time‐Periodic Systems

Benner

Denißen

Kohaupt

2013

Proc Appl Math and Mech

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Linear time-periodic systems have been an active area of research in the last decades. They arise in various applications such as anisotropic rotor-bearing systems and nonlinear systems linearized about a periodic trajectory. Rigorous bounds support the transient analysis of these systems. Optimal constants are determined by the differential calculus for norms of matrix functions. Bounds based on trigonometric spline approximations of the solution are introduced and convergence results for the approximations are stated. Bounds are illustrated by means of an anisotropic rotor-bearing system. Linear time-periodic systemsWe are considering a set of linear ordinary differential equations (ODEs) with time-periodic coefficients with periodicity T and a given initial condition as a linear time-periodic systeṁwhere x ∈ R n and A : R → R n×n . Under well known assumptions a unique solution to (1) is guaranteed. The solution can then be represented by the following theorem due to Floquet [1], but in a different form.Theorem 1.1 A fundamental matrix Φ(t) of (1) can be represented aswhere R, Z(t) ∈ C n×n and Z(t) = Z(t + T ) is nonsingular ∀t ∈ R.The solution x(t) is a linear combination of n linearly independent solutions w.r.t. the initial condition x 0 : x(t) = Φ(t)x 0 = Z(t)e Rt x 0 with Φ(0) = I. Determining Φ(t) in the interval [0, T ] is sufficient due to the semigroup property of the solution. For the sake of simplicity we consider equidistant nodes t i = ih for i = −1, 0, 1, . . . , r with h = T r and we use quadratic trigonometric splines T i (t) [2] to approximate the solution. . .whereT denotes a vector of the i-th coefficients α (j)i of the spline approximation s j (t) for j = 1, . . . , n. Quadratic trigonometric splines with compact support are chosen in order to simplify the computation of the coefficientswhere θ = 1 sin h sin h 2 . Solving ODEs by spline approximations has been investigated by F. R. Loscalzo and T. D. Talbot [3] and many others, e.g. A. Nikolis [4]. The focus on trigonometric splines is due to the periodicity of (1) and furthermore, we will equip the computation with rigorous upper and lower bounds on the solution. Demanding that the approximation s fulfills (1) at the nodes t i , yields a series of r + 1 linear systems A

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Solution formulas for differential Sylvester and Lyapunov equations

Behr

Benner

Heiland

2019

Calcolo

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The differential Sylvester equation and its symmetric version, the differential Lyapunov equation, appear in different fields of applied mathematics like control theory, system theory, and model order reduction. The few available straight-forward numerical approaches if applied to largescale systems come with prohibitively large storage requirements. This shortage motivates us to summarize and explore existing solution formulas for these equations. We develop a unifying approach based on the spectral theorem for normal operators like the Sylvester operator S(X) = AX + XB and derive a formula for its norm using an induced operator norm based on the spectrum of A and B. In view of numerical approximations, we propose an algorithm that identifies a suitable Krylov subspace using Taylor series and use a projection to approximate the solution. Numerical results for large-scale differential Lyapunov equations are presented in the last sections.

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Solution of the matrix eigenvalue problem VA+A*V=μV with applications to the study of free linear dynamical systems

Cited by 13 publications

References 26 publications

Rayleigh-Quotient Representation of the Real Parts, Imaginary Parts, and Moduli of the Eigenvalues of General Matrices

Rayleigh-Quotient Representation of the Real Parts, Imaginary Parts, and Moduli of the Eigenvalues of General Matrices

Bounds on the Solution of Linear Time‐Periodic Systems

Solution formulas for differential Sylvester and Lyapunov equations

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