Eigenvalue analysis determinant

The stability characteristics of the steady-state points is then determined via an eigenvalue analysis of the linearized version of the two DE (7.198) and (7.199). The linearized form of equations (7.198) and (7.199) is as follows ... 

Stability is determined by the eigenvalue analysis at an equilibrium point for flows and by the characteristic multiplier analysis of a periodic solution at a fixed point for maps [3]. 

The reduced correlation matrix is subsequently subjected to a PCA the eigenvalues are determined and normahzed to length 1, as explained in Example 5.3. The significant eigenvectors then determine the loading matrix i. This approach is termed principal factor analysis. 

An eigenvalue analysis is performed after every 5 increments to determine the residual stability of the system. Buckling can be detected as the moment when the first vibration frequency of the system becomes zero. 

Singularity of the matrix A occurs when one or more of the eigenvalues are zero, such as occurs if linear dependences exist between the p rows or columns of A. From the geometrical interpretation it can be readily seen that the determinant of a singular matrix must be zero and that under this condition, the volume of the pattern P" has collapsed along one or more dimensions of SP. Applications of eigenvalue decomposition of dispersion matrices are discussed in more detail in Chapter 31 from the perspective of data analysis. 

The PLS algorithm is relatively fast because it only involves simple matrix multiplications. Eigenvalue/eigenvector analysis or matrix inversions are not needed. The determination of how many factors to take is a major decision. Just as for the other methods the right number of components can be determined by assessing the predictive ability of models of increasing dimensionality. This is more fully discussed in Section 36.5 on validation. 

The basic idea is very simple In many scenarios the construction of an explicit kinetic model of a metabolic pathway is not necessary. For example, as detailed in Section IX, to determine under which conditions a steady state loses its stability, only a local linear approximation of the system at this respective state is needed, that is, we only need to know the eigenvalues of the associated Jacobian matrix. Similar, a large number of other dynamic properties, including control coefficients or time-scale analysis, are accessible solely based on a local linear description of the system. 

Then using these 91 peaks only, the original data set was reexamined by principal components analysis. Eigenvalues greater than one were plotted to determine how many factors should be retained. After variraax rotation, the factor scores were plotted and interpreted. 

The local stability and character of the singular points can be determined by the usual analysis of the eigenvalues of the Jacobian matrix... 

The Smith-Ewart equations can be solved using a single numerical eigenvalue determination under all conditions. Analytical solutions can also be obtained if n is not too large (n< 0.7)(8,9,10). These solutions encompass both the steady state and the approach to the steady state. Thus the particle number concentrations N, N, . .are known once P, k and c have been determined experimentally. As will be seen, these populations are the starting point for the MWD analysis. 

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