Non-singular

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... 

Equation (8.90) is non-singular since it has a non-zero determinant. Also the two row and column vectors can be seen to be linearly independent, so it is of rank 2 and therefore the system is controllable. 

N is of rank 2 and therefore non-singular, henee the system is eompletely observable and the ealeulation of an appropriate observer gain matrix Kg realizable. 

An nxp matrix X with n>p is called singular if linear dependences exist between the columns of X, otherwise the matrix is called non-singular. In this case the rank of X equals p minus the number of linear dependences among the columns of X. If n < p, then X is singular if linear dependences exist between the rows of X, otherwise X is non-singular. In that case, the rank of X equals n minus the number of linear dependences among the rows of X. A matrix is said to be of full rank when X is non-singular or alternatively when riX) equals the smaller of n or p. 

If the determinant A of the matrix A vanishes, then the matrix A is said to be singular. Otherwise, the matrix A is non-singular. 

It follows from equation (1.27) that the product of two non-singular matrices is also non-singular. 

Any non-singular square matrix A possesses an inverse matrix A defined as... 

If A is non-singular, then in matrix notation the vector x is related to the vector y by... 

The matrix X is easily seen to be unitary. Since the n eigenvectors are linearly independent, the matrix X is non-singular and its inverse X exists. If we multiply equation (1.57) from the left by X , we obtain... 

It is a common problem to solve a set of homogeneous equations of the form Ax = 0. If the matrix is non-singular the only solutions are the trivial ones, x = x2 = = xn = 0. It follows that the set of homogeneous equations has non-trivial solutions only if A = 0. This means that the matrix has no inverse and a new strategy is required in order to get a solution. 

Finally, since X is non-singular its inverse X-1 exists and premultiplication by X-1 yields the desired result... 

If P and Q are non-singular matrices, then A and B are said to be equivalent when B = PAQ. An important special case arises when PQ = 7, when... 

In an n-dimensional space L, the linear operators of the representation can be described by their matrix representatives. This procedure produces a homomorphic mapping of the group G on a group of n x n matrices D(G), i.e., a matrix representation of the group G. From equations (6) it follows that the matrices are non-singular, and that... 

The matrix gab is symmetric and non-singular and has other purely algebraic properties that can be demonstrated by first defining the matrices... 

As discussed in Pope (1997), a non-singular scaling matrix B can be introduced such that 0 < 5 JVAETBTBEVTa 0 < E oX defines the EOA. 

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.

We assume that we have shifted to a real new vector base, i.e., the vectors Sj are independent and B is non-singular. Now, we can go back to the v coordinates and write that, in a Euclidian space, vector length is an invariant... 

Let s assume the elements ci, a and cz of vector c are the unknowns. Thus, the system is comprised of three equations with three unknowns. Such systems of n equations with n unknowns have exactly one solution if none of the individual equations can be expressed by linear combinations of the remaining ones, i.e. if they are linearly independent. Then, the coefficient matrix A is of full rank and non-singular and its inverse, A1, exists such that right multiplication of equation (2.20) with A 1 allows the determination of the unknowns. 

S. G. Larsson and A. J. Carlsson, Influence of Non-Singular Stress Terms and Specimen Geometry on Small Scale Yielding at Crack Tips in Elastic-Plastic Material, J. Mech. Phys. Solids, 21, 263-278 (1973). 

The limit c —can now be taken provided that (1) V is everywhere non-singular, which is true for finite nuclei [42] but not point nuclei, and that (2) E < c which is true for the (shifted) positive-energy solutions only. With this procedure all relativistic effects are eliminated and one obtains the four-component non-relativistic Levy-Leblond equation [34,43]... 

Irrespective of the sources of phenolic compounds in soil, adsorption and desorption from soil colloids will determine their solution-phase concentration. Both processes are described by the same mathematical models, but they are not necessarily completely reversible. Complete reversibility refers to singular adsorption-desorption, an equilibrium in which the adsorbate is fully desorbed, with release as easy as retention. In non-singular adsorption-desorption equilibria, the release of the adsorbate may involve a different mechanism requiring a higher activation energy, resulting in different reaction kinetics and desorption coefficients. This phenomenon is commonly observed with pesticides (41, 42). An acute need exists for experimental data on the adsorption, desorption, and equilibria for phenolic compounds to properly assess their environmental chemistry in soil. 

Obviously, Horiuti numbers are defined up to non-singular linear transformation. 

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