Singular, matrix

Let us express the displacement coordinates as linear combinations of a set of new coordinates y >q= Uy then AE = y U HUy. U can be an arbitrary non-singular matrix, and thus can be chosen to diagonalize the synmietric matrix H U HU = A, where the diagonal matrix A contains the (real) eigenvalues of H. In this fomi, the energy change from the stationary point is simply AF. = t Uj A 7- h is clear now that a sufBcient... 

Singular matrix A matrix is singular if its determinant is zero. 

If no laminae have failed, the load must be determined at which the first lamina fails (so-called first-ply failure), that is, violates the lamina failure criterion. In the process of this determination, the laminae stresses must be found as a function of the unknown magnitude of loads first in the laminate coordinates and then in the principal material directions. The proportions of load (i.e., the ratios of to Ny, to My,/ etc.) are, of course, specified at the beginning of the analysik The loaa parameter is increased until some individual lamina fails. The properties, of the failed lamina are then degraded in one of two ways (1) totally to zero if the fibers in the lamina fail or (2) to fiber-direction properties if the failure is by cracking parallel to the fibers (matrix failure). Actually, because of the matrix manipulations involved in the analysis, the failed lamina properties must not be zero, but rather effectively zero values in order to avoid a singular matrix that could not be inverted in the structural analysis problem. The laminate strains are calculated from the known load and the stiffnesses prior to failure of a lamina. The laminate deformations just after failure of a lamina are discussed later. 

It should be noted that in the case of a singular matrix A, the dimensions of V and A are pxr and rxr, respectively, where r is smaller than p. The expression in eq. (29.53) allows us to compute the generalized inverse, specifically the Moore-Penrose inverse, of a symmetric matrix A from the expression ... 

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. 

Elements of the matrix —(ft2/2p)W i are usually small in the vicinity of a conical intersection and can be added to zd to give a corrected diabatic energy matrix. As can be seen, whereas in Eq. (15) W ad contains both the singular matrix W ad and the nonsingular one W ad, Eq. (31) contains only the latter. Nevertheless, the residual first-derivative coupling term w ad Vr does not vanish. 

This result can be generalized into the statement that any arbitrary vector in n dimensions can always be expressed as a linear combination of re basic vectors, provided these are linearly independent. It will be shown that the latent solutions of a singular matrix provide an acceptable set of basis vectors, just like the eigen-solutions of certain differential equations provide an acceptable set of basis functions. 

Note that the denominator of (A. 17), the determinant of A = A, is a scalar. If A = 0, the inverse does not exist. A square matrix with determinant equal to zero is called a singular matrix. Conversely, for a nonsingular matrix A, det A 0. 

Figure 2.1 Hyper-prism built on the column-vectors (alf a2, a3) of a non-singular matrix A3 x 3. The determinant is equal to the volume of the hyper-prism. When one of the vectors can be expressed as a linear combination of the others, both the volume and the determinant vanish and the matrix A is singular.

Equations (4.40) and (4.41) are easily implemented in an existing least-squares program and give both the constrained and the unconstrained results in a single refinement cycle. However, the method fails if the unconstrained refinement corresponds to a singular matrix, as would be the case, for example, if all population parameters, including those of the core functions, were to be refined in addition to the scale factor k. 

Our equivalent SPICE model of this circuit is shown in Fig. 9.2. A 100 M 2 resistor (R4) has been added from the inverting terminal of the current sense pin to ground in order to help convergence and prevent singular matrix errors. 

The trick described above is a good way to prevent singular matrix errors. Singular matrix errors usually occur because the node does not have a direct path to ground. When the 100 resistor is added, this node has a direct path to ground and does not affect the operation of the circuit. 

SUCCESSFUL DECOMPOSITION 1 SINGULAR MATRIX MATRIX FACTORS IN PACKED FORM... 

Notice that when we multiply eqn (4-3.10) by B-1 to produce eqn (4-3.11), as matrices do not necessarily commute, we must do so on the right-hand side of both sides of the equation. A method for finding the inverse of a non-singular matrix is given in Appendix A.4-2. From this method it is apparent why A-1 is only defined when det(A) = 0. 

Note specifically the differential operator function function dydx which is central to the program, as well as the boundary condition settings in function Rand, the singularity matrix S, and the various starting solution guesses at the end of the program. 

If r r (cases (a) and (c)) then there exits a non-singular matrix Z such that... 

Recall that if Ly forms a basis for T, then even number. If T is not F, case (b), v and ipr are linearly independent (LI) and so time reversal causes a doubling of degeneracy. If V F cases (a) and (c), then there exists a non-singular matrix Z which transforms T into P ... 

Two matrix systems of the same order g arc equivalent if there exists a non-singular matrix S such that... 

Now suppose that T1 TJ and let Z be the non-singular matrix that transforms TJ into T ... 

Their mutual Poisson brackets calculated using (34) yield a non-singular matrix with elements... 

Let us assume that there exists a non-singular matrix A which transforms basis x into orthonormal basis set y,... 

M is a singular Matrix. Zero entries on the main diagonal of this matrix identify the algebraic equations, and all other entries which have the value 1 represent the differential equation. The vector x describes the state of the system. As numeric tools for the solution of the DAE system, MATLAB with the solver odel5s was used. In this solver, a Runge Kutta procedure is coupled with a BDF procedure (Backward Difference Formula). An implicit numeric scheme is used by the solver. 

The isotopic differences need not be taken as differences with respect to the parent, Pg(s) - /y 1). Any arrangement, Pg(s") - Pg(s ), which can be transformed by a non-singular matrix C into the former set will result in the same restructure. Occasionally, rQ-type fits have been reported where the isotopic differences Pg(s) -Pg 1) for the isotopomers s = 2, Ns,g=x,y,z, plus the three moments of the parent, /yi), were fitted, in the hope of compensating for part of the rovib contamination of the moments. However, the simple linear transformation between this combination of observations and the complete set of individual moments P (1),.., Pg(s),. 

We can clearly see from equation (3.99) how regularization makes the nearly singular matrix well-conditioned even if Qi —+ 0 division by zero does not occur. 

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