Matrix 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... 

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.

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. 

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,... 

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),. 

In fact these are just particular instances of a similarity transform, in which the matrix is pre-multiplied by a non-singular matrix and post-multiplied by its inverse... 

Equation (4) implies that (detA) = 1, detA = 1. It follows that A is a non-singular matrix, with A = g Ag. Along with contravariant vectors we associate covariant vectors (covectors) a such that... 

Mathematically, the relative gain array (RGA) of a non-singular matrix G is defined as ... 

A representation is called reducible if its matrices can be transformed into the so-called block form by using the transformation P r(Ri)P for every matrix F(Ri), where P is a non-singular matrix. 

In this form, E is a, non-singular matrix, and E and F have non-negative entries. The outer iterations of (5.2) can now be rephrased as... 

Contrarily to what happens with PARAFAC, TuckerS models are not unique, as rotational freedom can exist. Indeed, if the model described in Equation (11) is replaced with another one where the loadings along one mode (let us say the first one, for the sake of simplicity) are transformed by multiplication by a square non-singular matrix R, and the core is counter-rotated by the same fit is obtained ... 

The matrix C is a non-singular matrix of dimensions (3N-3)><3N with elements +1, -1 and 0 appropriately placed to define the coimection between the coordinate X and Xg. This has the effect that the elements of D are numerically equivalent to the elements of Px less the polar tensor of the first atom referring to the colunm array Xg. 

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