Eigenvalues complex

The set of roots of a polynomial with real coefficients can include conjugate pairs of complex numbers. Thus eigenvalues can be complex, appearing in conjugate pairs. When this happens the corresponding column eigenvectors also form a conjugate pair, as do the rows. 

If we wish to remain in the real domain, this can be done by observing that the product 

Although the first form has fewer off-diagonal non-zeroes, the second form has only real elements, in both eigenvalues and eigenvectors, ft also leads to some intuitive understanding, that if we multiply both eigencolumns by M the result is a pair of vectors rotated in the space spanned by the two eigencolumns, as well as scaled. 

Eigenfactorisation of a matrix helps us to understand what happens when vectors are multiplied by the matrix repeatedly. 

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Solving the time-dependent Sclirodinger equation for resonance states [78] one obtains a set of complex eigenvalues, which may be written in the fonn... 

Finally, the simplest approach to extract resonances is to add to the Hamiltonian an absorbing potential [8, 48. 108. 109], and then look for the complex eigenvalues of the Hamiltonian /7-i Vj. The absorbing potential... 

The curious shape of the computed parts of the curves suggests that there is, in each case, a discontinuity of slope. However, examination of the results shows that there is, in fact, a switch in the dominant eigenvalue as the Lewis number changes. Above a certain value of the Lewis number this is real and moves to the right as decreases. But eventually it crosses with a pair of complex eigenvalues moving to the left and these, which become the dominant eigenvalues for smaller values of, cause... 

A.7 Show that for a 2 X 2 symmetrical matrix, the eigenvalues must be real (do not contain imaginary components). Develop a 2 X 2 nonsymmetrical matrix which has complex eigenvalues. 

Calculate the eigenvalues of. If the system is N x N (JV controlled variables and N manipulated variables) there will be N complex eigenvalues. The IMSL subroutine EIGCC is used in the program given in Table 16.2. 

The values of Km and T2d from Eq.(36) can be obtained from the transfer function of the linearized model at the equilibrium point, applying conventional methods from the linear control theory (see [1]). In order to investigate the self-oscillating behavior, one can determine the linearized system at the equilibrium point, and the corresponding complex eigenvalues with zero real part, when the parameters Km and of the PI controller are varied. For example, taking into account Eq.(34), the Jacobian matrix of the linearized system at dimensionless set point temperature xs is the following ... 

Consider the space state model R deflned by Eq.(52), showing an equilibrium point such that the matrix of the linearized system at this point has a real negative eigenvalue A and a pair of complex eigenvalues a j/3, j = /—1) with positive real parts 0.. In this situation, the equilibrium point has onedimensional stable manifold and two-dimensional unstable manifold. If the condition A < a is verified, it is possible that an homoclinic orbit appears, which tends to the equilibrium point. This orbit is very singular, and then the Shilnikov theorem asserts that every neighborhood of the homoclinic orbit contains infinite number of unstable periodic orbits. 

In two-parametric families three constants can meet. If three smallest constants kj,ki and L have comparable values and are much smaller than others, then static and dynamic properties would be determined by these three constants. Stationary rate w and dynamic of relaxation for the whole cycle would be the same as for 3-reaction cycle A B C A with constants kj,ki and km- The damped oscillation here are possible, for example, if kj — ki—km—k, then there are complex eigenvalues X— k(—(3/1) 1( /3/ )). Therefore, if a cycle manifests damped oscillation, then at least three slowest constants are of the same order. The same is true, of course, for more general reaction networks. 

Fig. 9. Linear stability diagram illustrating (21). The bifurcation parameter B is plotted against the wave number n. (a) and (b) are regions of complex eigenvalues < > (b) shows region corresponding to an unstable focus (c) region corresponding to a saddle point. The vertical lines indicate the allowed discrete values of n. A = 2, D, = 8 103 D2 = 1.6 10-3.

Again 2, and k2 are complex eigenvalues of the form Re(x) + ilm(A), but now they have positive real parts. Thus the exponential terms in (3.47) and (3.48) grow in time. The perturbation grows away from the unstable stationary state in a divergent oscillatory or unstable focal manner. 

In fact only the upper root corresponds to a Hopf bifurcation point (the lower solution to the condition tr(J) = 0 being satisfied along the saddle point branch of the isola where the system does not have complex eigenvalues). 

For convenience we will make a simple demonstration of how to transform a 2x2 matrix problem to complex symmetric form. In so doing we will also recognise the appearence of a Jordan block off the real axis as an immediate consequence of the generalisation. The example referred to is treated in some detail in Ref. [15], where in addition to the presence of complex eigenvalues one also demonstrates the crossing relations on and off the real axis. The Hamiltonian... 

Some real matrix classes, studied in subsection (F) below, however, have only real eigenvalues and corresponding real eigenvectors. The complication with complex eigenvalues and eigenvectors is caused by the Fundamental Theorem of Algebra which states that all the roots of both real and complex polynomials can only be found in the complex plane C. 

P. Froelich, E.R. Davidson, E. Brandas, Error Estimates for Complex Eigenvalues of Dilated Schrodinger Operators, Phys. Rev. A28 (1983) 2641. 

M. Hehenberger, P. Froelich, E. Brandas, Weyl s Theory Applied to Predissociation by Rotation. II. Determination of Resonances and Complex Eigenvalues Application to HgH, J. Chem. Phys. 65 (1976) 4571. 

Despite being called a state, a resonance does not show up as an eigenstate of an Hermitian Hamiltonian. However, as it represents a particle state that is localized in space for some time and that delocalizes with a small but finite rate, a resonance is reminiscent of a stationary state, but with a decaying norm. Indeed, it can be shown that we can represent resonance states as eigenstates of a non-Hermitian Hamiltonian, whose complex eigenvalues lie in the lower half of the complex plane. 

Complex rotation can be usefully applied also to the case of the interaction of an atom with a time-dependent perturbation. With the Floquet formalism by Shirley [41], it was shown that, for a time-periodic field, the dressed states of the combined atom-field system can be characterized non-perturbatively by the eigenstates of a time-independent, infinite-dimensional matrix. The combination of the Floquet approach with complex rotation, proposed by Chu, Reinhardt, and coworkers [37, 42, 43], permits to account for the field-induced coupling to the continuum in an efficient way. As in the time-independent case, this results in complex eigenvalues (this time to the Floquet Hamiltonian matrix) and again the imaginary parts give the transition rate to the continuum. This combination has since then been successfully used to examine various strong field phenomena a review can be found in Ref. [44]. 

C. A. Nicolaides, Th. Mercouris, Partial widths and interchannel coupling in autoionizing states in terms of complex eigenvalues and complex coordinates, Phys. Rev A 32 (1985) 3247. 

At last, perform a shift in the imaginary part of the complex eigenvalues in the same spirit as above, in order to make real the ground-state energy too corresponding to v = 0. This leads us to write... 

The complex eigenvalues of F are also poles of t, and therefore they are solutions of the equations... 

This model is a special case of the model studied by Matis and Wehrly [369] in which Ai Erl(Ai, u ) and A2 Ev A2.V2) retention-time distributions are associated with the first and second compartments, respectively. The analysis of the characteristic polynomial of this model implies that there are at least two complex eigenvalues, except for the case v = 2 with parameters satisfying the condition... 

The steep negative slope

operating point in such cases is a stable focus. In contrast, shallow negative slopes... 

The form of this correction can be appreciated by comparison with the expression for the /t-doubling terms themselves, equation (8.400). There is, however, a problem with this form for the Hamiltonian operator because the two operator factors, such as (J2 + J2) and (J — S )2 do not commute with each other. The Hamiltonian (8.421) is therefore not Hermitian and so has complex eigenvalues. The operator can be made to have Hermitian form by taking the so-called Hermitian average,... 

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