Eigenvalue analysis similar matrices

Q-mode factor analysis is based on a major product matrix, XX. Whereas the R-mode analyses focus on interrelationships among variables, Q-mode analyses focus on interrelationships among objects. Accordingly, the major product matrix is usually a distance or similarity matrix. Formally, Q-mode and R-mode factor analyses are closely related because the nonzero eigenvalues of the major product matrix are identical to the eigenvalues of the minor product matrix, and the eigenvectors are easily derived from one another (28). 

Then, a hash string, describing the molecule, is obtained by concatenation of the highest and lowest eigenvalues of the matrix Q expressed to six decimal places. This string representation of compounds is suitable for similarity/diversity analysis. 

On the surface, factor analysis and principal component analysis are very similar. Both rely on an eigenvalue analysis of the covariance matrix, and both use linear combinations of variables to explain a set of observations. However, in PCA the quantities of interest are the observed variables themselves the combination of these variables is simply a means for simplifying their analysis and interpretation. Conversely, in factor analysis the observed variables are of little intrinsic value what is of interest is the underlying factors. 

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. 

In view of the Hessian character (10.20) of the thermodynamic metric matrix M(c+2), the eigenvalue problem for M(c+2) [(10.23)] can be usefully analogized with normal-mode analysis of molecular vibrations [E. B. Wilson, Jr, J. C. Decius, and P. C. Cross. Molecular Vibrations (McGraw-Hill, New York, 1955)]. The latter theory starts from a similar Hessian-type matrix, based on second derivatives of the mechanical potential energy Vpot (cf. Sidebar 2.8) rather than the thermodynamic internal energy U. 

Erom these Eudidean-distance matrix, M/M quotient matrix and L/L quotient matrix, —> leading eigenvalues were calculated to perform similarity/diversity analysis. 

RDA modifies (shrinks) the eigenvalues of the original covariance matrices. Other methods exist with similar goals. Amongst these is DASCO (discriminant analysis with shrunken covariances). Although popular with the chemometric community, DASCO is quite complex, yet does not outperform RDA or the simpler ridge method. The latter also modifies the sample covariance matrix to be used with LDA from S to S + a I, where a is a parameter to be optimized, I the unit matrix. 

Golub GH (1973) Some modified matrix eigenvalue problems. SIAM Rev 15 318-334 Gordon RG (1968) Error bounds in equilibrium statistical mechanics. J Math Phys 9 655-672 Gottlieb D, Orszag SA (1977) Numerical analysis of spectral methods. SIAM, Philadelphia Grace JR, Taghipour F (2004) Verification and validation of CFD models and dynamic similarity for fluidized beds. Powder Technol 139 99-110... 

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