Nonsingular matrix coordinates

Nodes, line of, 280 Nondegenerate species (see Species) Nonsingular matrix, 205 Normal coordinates, 19 characters of, 94 degenerate, 93, 107... 

Such transformations (and their matrices) are called nonsingular. A nonsingular matrix is recognizable through the fact that its determinant must be nonzero. If we had a transformation which, for example, caused all or some points in one coordinate system to coalesce into a single point in the transformed system, we would lose our ability to back-transform in a unique way. Such a singular transformation has no inverse, and the determinant of a singular matrix equals zero. 

Here, is the mobility tensor in the chosen system of coordinates, which is a constrained mobility for a constrained system and an unconstrained mobility for an unconstrained system. As discussed in Section VII, in the case of a constrained system, Eq. (2.344) may be applied either to the drift velocities for the / soft coordinates, for which is a nonsingular / x / matrix, or to the drift velocities for a set of 3N unconstrained generalized or Cartesian coordinates, for a probability distribution (X) that is dynamically constrained to the constraint surface, for which is a singular 3N x 3N matrix. The equilibrium distribution is. (X) oc for unconstrained systems and... 

Following a similar procedure to the one employed above, it is easy to verify that we obtain a model that approximates the fast dynamics of the system in Figure 4.2, in the form of Equation (4.20). Also, it can be verified that only 2N + 8 of the 2N + 9 steady-state constraints that correspond to the fast dynamics are independent. After controlling the reactor holdup Mr, the distillate holdup Md, and the reboiler holdup MB with proportional controllers using respectively F, D, and B as manipulated inputs, the matrix Lb (x) is nonsingular, and hence the coordinate change... 

Equation (18) is easily solved for X if the matrix t WD is nonsingular. If, however, the moments are not sensitive to one or more of the coordinates, or if two or more coordinates are nearly linearly dependent,... 

For the condition given in Eq. 2.6 the matrix in Eq. 2.4 is said to be nonaingxdar. Alternatively, the matrix of Eq. 2.4 is nonsingular if there is another matrix called the inverse of the matrix in Eq. 2.4 such that the successive application of transformation 2.4 and its inverse brings the system into its original form. The successive application of two transformations must be equivalent to one single transformation which will have its own matrix. Consider the transformation shown in Eq. 2.5 which expresses the p in terms of the p s and then the transformation in Eq. 2.7 which takes the p in a new system of coordinates p" with coeflScients b. 

This result is the motivation behind the use of a preconditioner, a transformation of the linear system into a related one whose condition number is closer to 1. This reduces the number of iterations necessary for iterative methods to converge to a solution, and is common practice in the numerical solution of BVPs. Let us choose some nonsingular upper-triangular matrix Mi defining a coordinate transformation,... 

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