Singular eigenvalues

MSN. 138. T. Petrosky and I. Prigogine, Complex spectral representations and singular eigenvalue problem of Liouvillian in quantum scatering, in Symposium Quantum Physics and the Universe, Waseda University, Tokyo, Japan, 1992. 

Let u be a vector valued stochastic variable with dimension D x 1 and with covariance matrix Ru of size D x D. The key idea is to linearly transform all observation vectors, u , to new variables, z = W Uy, and then solve the optimization problem (1) where we replace u, by z . We choose the transformation so that the covariance matrix of z is diagonal and (more importantly) none if its eigenvalues are too close to zero. (Loosely speaking, the eigenvalues close to zero are those that are responsible for the large variance of the OLS-solution). In order to liiid the desired transformation, a singular value decomposition of /f is performed yielding... 

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... 

The singular values of a complex n x m matrix A, denoted by cr,(A) are the nonnegative square-roots of the eigenvalues of A A ordered such that... 

For zero field of view (Fig. 8), as expeeted, there are respeetively 2, 4 and 8 singular modes when the piston and the tilts are measured from the LGSs, when the piston is not but the tilt is (ease of the polychromatie LGS, see 15.3), and when neither the piston nor the tilts are measured (monoehromatie LGS ease). Even and odd modes eorrespond respeetively to the high and low eigenvalues (lowest and highest modes). 

When the piston is assumed to be measured from the LGSs (top eurve), the corresponding mode is singular because one cannot measure the contribution of each layer to the total piston. This case is not realistic, since no wavefront sensor measures the piston. When the tilts are measured from the LGSs (case of the polychromatic LGS), the odd piston is not measured again. The even piston is no longer available. And the two odd tilt modes are not also, because whereas the tilt is measured, the differential tilt between the two DMs is not one does not know where the tilt forms. Thus there are 4 zero eigenvalues. 

For some isomerizing species, such as LiCN/LiNC [9], quantum monodromy may be demonstrated by transporting a unit cell around a finite fold of the Em map, which joins the interleaving eigenvalues of the separate isomers. This fold therefore plays the role of the simple focus-focus singularity. 

Figure 22. Quantum eigenvalues (dots) plotted against angular momentum, for two different slices through Fig. 21. Left, n —1 = 22 right. 3n + / = 80. The circle in each panel is the intersection with the singular thread, which lies at L = 0. Taken from Ref. [13] with permission of the American Institute of Physics, Copyright 2004.

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. 

Fig. 31.13. Schematic example of three common algorithms for singular value and eigenvalue decomposition.

Correspondence factor analysis can be described in three steps. First, one applies a transformation to the data which involves one of the three types of closure that have been described in the previous section. This step also defines two vectors of weight coefficients, one for each of the two dual spaces. The second step comprises a generalization of the usual singular value decomposition (SVD) or eigenvalue decomposition (EVD) to the case of weighted metrics. In the third and last step, one constructs a biplot for the geometrical representation of the rows and columns in a low-dimensional space of latent vectors. 

Both types of symmetric displays exhibited in Figs. 32.9 and 32.10 have their merits. They are called symmetric because they produce equal variances in the scores and in the loadings. In the case when a = 3 = 1, we obtain that the variances along the horizontal and vertical axes are equal to the eigenvalues h associated to the dominant latent vectors. In the other case when a = P = 0.5, the variances are found to be equal to the singular values X. 

The condition number is always greater than one and it represents the maximum amplification of the errors in the right hand side in the solution vector. The condition number is also equal to the square root of the ratio of the largest to the smallest singular value of A. In parameter estimation applications. A is a positive definite symmetric matrix and hence, the cond ) is also equal to the ratio of the largest to the smallest eigenvalue of A, i.e.,... 

S = 29.5803 0 0 0 1.9907 0 0 0 0.2038 Display the S matrix or the singular values matrix. This diagonal matrix contains the variance described by each principal component. Note the squares of the singular values are termed the eigenvalues. 

V = 0.2380 -0.9312 0.2762 0.6279 -0.0694 -0.7752 0.7410 0.3579 0.5681 Display the V matrix or the right singular values matrix this is also known as the loadings matrix. Note this matrix is the eigenvectors corresponding to the positive eigenvalues. 

Due to the conservation of elements, the rank of J will lie less than or equal to K — E 1 In general, rank(J) = Ny < K - E, which implies that V = K — T eigenvalues of J are null. Moreover, since M is a similarity transformation, (5.51) implies that the eigenvalues of J and those of J are identical. We can thus limit the definition of the chemical time scales to include only the Nr finite ra found from (5.50). The other N components of the transformed composition vector correspond to conserved scalars for which no chemical-source-term closure is required. The same comments would apply if the Nr non-zero singular values of J were used to define the chemical time scales. 

A symmetric matrix A, can usually be factored using the common-dimension expansion of the matrix product (Section 2.1.3). This is known as the singular value decomposition (SVD) of the matrix A. Let A, and u, be a pair of associated eigenvalues and eigenvectors. Then equation (2.3.9) can be rewritten, using equation (2.1.21)... 

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