Matrix characteristic

Thus the characteristic matrix for this closedloop system is the 4cl niatrix. Its eigenvalues will be the close oop eigenvalues, and they will be the roots of the closedloop characteristic equation. 

For each layer zy isemi-infinite space z>Zn there is a characteristic matrix M. The latter can be used to relate the tangential fields at the first phase boundary at z = zi = 0 to those at the final phase boundary at z = Z/v i by use of the following equation ... 

The Fresnel reflection coefficients r for the overall system can be obtained from the elements m of the characteristic matrix M associated with the overall system ... 

Beyond the simple thin surfactant monolayer, the reflectivity can be interpreted in terms of the internal structure of the layer, and can be used to determine thicker layers and more complex surface structures, and this can be done in two different ways. The first of these uses the optical matrix method [18, 19] developed for thin optical films, and relies on a model of the surface structure being described by a series or stack of thin layers. This assumes that in optical terms, an application of Maxwell s equations and the relationship between the electric vectors in successive layer leads to a characteristic matrix per layer, such that... 

A vector terminating rth of the distance between two characteristic vectors with zero characteristic roots The characteristic matrix for the unperturbed rate constant matrix ... 

The characteristic matrix for the perturbed rate constant matrix K... 

Each of the stmctmal parts (subsystems) of the system (4.10) have the characteristic matrix V.. ... 

Each structural part (subsystem) has a characteristic matrix of the following form ... 

The reflection and transmission coefficients for the whole stack can be extracted from the characteristic matrix of the whole stratified medium using the following relationships ... 

For this approach each layer in the interface as shown in figure 3.3 is represented by a characteristic matrix thus for layer m... 

The elementary characteristic matrix Mj for each of the — 2 constiment layers in the particular stratified medium is calculated as... 

The characteristic matrix of the whole multilayer structure is calculated as the product of the elementary matrices obtained at step 7 ... 

The layers in the interface (Figure 9.11) can be represented by a characteristic matrix of the form for an m-layered system ... 

Upon expansion, an 8th-order polynomial equation in A arises. Eight roots of the polynomial exist. Even if all the values in the characteristic matrix are real, some roots may be complex. When complex roots occur, they appear in pairs. The roots of the polynomial are called eigenvalues of the characteristic matrix. The polynomial equation is called the eigenvalue equation. 

FIGURE 12.1 Characteristic matrix for system of eight reactions in circle. 

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