Matrix of eigenvalues

Verify the result in Eq. (8-50) by finding the eigenvalues of matrix (8-49). What are the eigenvector coefficients ... 

Practical experience has shown that (i) if we have a relatively large number of data points, the prior has an insignificant effect on the parameter estimates (ii) if the parameter estimation problem is ill-posed, use of "prior" information has a stabilizing effect. As seen from Equation 8.48, all the eigenvalues of matrix A are increased by the addition of positive terms in its diagonal. It acts almost like Mar-quadt s modification as far as convergence characteristics are concerned. 

The eigenvalues of matrix (17) depend on the coefficients aij from the following equation ... 

For every monomolecular kinetic system, the Jordan cell for zero eigenvalue of matrix K is diagonal and the maximal number of independent linear conservation laws (i.e. the geometric multiplicity of the zero eigenvalue of the matrix K) is equal to the maximal number of disjoint ergodic components (minimal sinks). 

Assuming that all the absolute values of the eigenvalues of matrix S are less than one137, we can express the right hand side of Eq. (220) in a power series of S... 

The zeroth eigenvalues of matrix A and G are zero, so that without any approximation, one can write an equation for diffusion mode... 

Let us note, that the matrixes A and G are approximations of the real situation though, in any case, the zeroth eigenvalues of the matrixes must be zero and equation (4.1) for diffusive mode is valid, the other eigenvalues of matrix G depends, generally speaking, on the mode label. In fact, the written equations for the relaxation modes are implementation of the statements that the motion of a single macromolecule can be separated from others, and the motion of a single macromolecule can be expanded into an independent motion of modes. 

The roots uk f=1 of Eq. (103) coincide with the eigenvalues of matrix U. On the other hand, the function det[wl — U] is proportional to its Kth degree characteristic polynomial, which is equal to QK(u) up to an overall multiplicative constant of the normalization type ... 

The matrix eigenvalues are analyzed for the stationary state in respect to the concentrations of aU independent intermediates. In the thermody namic representation of kinetic equations, independent variables that describe the system evolution in time are not the concentrations but ther modynamic rushes A of the intermediates. Hence, the analysis of the sta bihty criterion in terms of thermodynamic variables needs an inspection of eigenvalues of matrix M p = However, the specific form of the... 

The number X of negative eigenvalues of matrix IH(rc) is called the index of the critical point rc. 

The set of eigenvalues is called the spectrum and is denoted as A (A). The maximum eigenvalue of matrix A is called the spectral radius of A ... 

The lowest three eigenvalues of matrix 51 differ only marginally from those of the adjacency matrix A (cf Figure 5, where + BA is shown) ... 

According to stability analysis of linear time invariant system, stability of the closed-loop system x= A- BK)x depends on the eigenvalue of eigenmatrix (A - BK). In other words, the condition that the stabifity is positive is all the eigenvalues of matrix (A - BK) are negative. The switching function of SMC is... 

It can be shown that the eigenvalues of matrix A are negative real numbers except the eigenvalue zero which corresponds to the equilibrium. The connection between the eigenvalues A, and lifetimes t, is as follows ... 

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