Nonsingular

This decomposition into a longitudinal and a hansverse part, as will be discussed in Section III, plays a crucial role in going to a diabatic representation in which this singularity is completely removed. In addition, the presence of the first derivative gradient term W l Rx) Vr x (Rx) in Eq. (15), even for a nonsingular Wi i (Rx) (e.g., for avoided intersections), introduces numerical inefficiencies in the solution of that equation. 

Stewart s argument provides a prescription for constructing a solution of equations (11.61) - (11.63) provided the matrix Is nonsingular for all relevant values of jc, and provided the differential equations (11.64) and (11.65) have solutions consistent with their boundary conditions. It is possible, in principle, to check the nonsingularity of for any... 

Looking at the matrix equation Ax = b, one would be tempted to divide both sides by matrix A to obtain the solution set x = b/A. Unfortunately, division by a matrix is not defined, but for some matrices, including nonsingular coefficient matrices, the inverse of A is defined. 

This relation is not in the best form for the calculation of the drag since 8 appears in both products. Hence it is necessary to change the two independent B-numbers by requiting that 8 occur in only one of them. To this end, we let Af be a nonsingular submatrix of of order 2 containing the tow corresponding to 8. Thus, tow 1 and, eg, tow 5 of ate chosen to give... 

Suppose that the problem is to find a B-matris of D such that the variables C, and E each occur in one and only one of the B-vectors. Since the submatris Af of Cconsisting of the first three rows corresponding to the variables C, and E is nonsingular, according to Theorem 6 there exists a B-matrix with the desired property. Let Af be the adjoint matrix of M. Then (eq. 52) ... 

Siace the columns of any complete B-matrix are a basis for the null space of the dimensional matrix, it follows that any two complete B-matrices are related by a nonsingular transformation. In other words, a complete B-matrix itself contains enough information as to which linear combiaations should be formed to obtain the optimized ones. Based on this observation, an efficient algorithm for the generation of an optimized complete B-matrix has been presented (22). No attempt is made here to demonstrate the algorithm. Instead, an example is being used to illustrate the results. 

If A is a square matrix and if principal submatrices of A are all nonsingular, then we may choose P as the identity in the preceding factorization and obtain A = LU. This factorization is unique if L is normahzed (as assumed previously), so that it has unit elements on the main diagonal. 

Since s and e are symmetric, = t-jiki = tyik- Since (5.3) has been assumed to be invertible in , c is nonsingular. Note that (5.4) may be written as... 

The remainder of this section will be concerned with a particular case in which normality conditions hold. The constitutive equation for the internal state variables (5.11) involves the constitutive function a, and the normality conditions (5.56) and (5.57) involve an unknown scalar factor y. In some circumstances, a may be eliminated and y may be evaluated by using the consistency condition. These circumstances arise if b is nonsingular so that the normality condition in strain space (5.56j) may be solved for k... 

Note that, since G and F are nonsingular, c is nonsingular. Similarly, if is nonsingular, b is also. Conversely, given the spatial stress rate relation (5.154)... 

Since the deformation tensor F is nonsingular, it may be decomposed uniquely into a proper orthogonal tensor R and a positive-definite symmetric tensor U by the polar decomposition theorem... 

Nonsingular matrix A matrix is nonsingular if its determinant is not zero. 

Table 5.4 FVaction of symmetric (0, l)-matrices which are nonsingular in finite fields T p (p-prime) - gives the probability that a random (undirected) lattice evolving according to an OT 0p rule yields reversible behavior.

Let D be a nonsingular diagonal matrix, and apply the theorem proved above to the matrix D — B. This matrix is nonsingular if and only if I — D 1B is nonsingular, and a sufficient condition for this is... 

If v = sup, the interpretation is that A = D — B is nonsingular, provided every diagonal dement exceeds in modulus the sum of the moduli... 

In the stationary methods, it is necessary that G be nonsingular and that p(M) < 1. In the methods of projection, however, Ca varies from step to step and is angular, while p(Ma) = 1. In these methods the vectors 8a are projected, one after another, upon subspaces, each time taking the projection as a correction to be added to xa to produce za+x- At each step the subspace, usually a single vector, must be different from the one before, and the subspaces must periodically span the entire space. Analytically, the method is to make each new residual smaller in some norm than the previous one. Such methods can be constructed yielding convergence for an arbitrary matrix, but they are most useful when the matrix A is positive definite and the norm is sff U. This will be sketched briefly. 

This provides an inductive, and a constructive, proof of the possibility of a triangular factorization of the specified form, provided only certain submatrices are nonsingular. For suppose first, that Au is a scalar, A12 a row vector, and A21 a column vector, and let Ln = 1. Then i u = A1U B12 — A12, and L2l and A22 axe uniquely defined, provided only Au = 0. But Au can be made 0, at least after certain row permutations have been made. Hence the problem of factoring the matrix A of order n, has been reduced to the factorization of the matrix A22 of order n — 1. 

Before this is done, however, a certain paradox needs to be discussed briefly. Given a matrix A, and a nonsingular matrix V, it is known that A, and V XA V, have the same characteristic polynomial, and the two matrices are said to be similar. Among all matrices similar to a given matrix A, there are matrices of the form... 

This is possible since P has maximal rank p. Now all roots of M are roots of A. Assume A to be nonsingular, and suppose that if A has any multiple roots occurring among the roots of M, then it has the same multiplicity as a root of M. Then it can be shown that... 

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