Matrices eigenvalues/eigenvectors

In order to find extrema of E( ui ), subject to the normalization condition, standard moves known as the Lagrange multipliers method are applied, which readily lead us to the well-known form of the generalized matrix eigenvalue/eigenvector problem ... 

In practice, the matrix (iL+R+K) is diagonalized first, with a matrix of eigenvectors, U, as in equation (B2.4.15)), to give a diagonal matrix. A, with the eigenvalues, X, of L down the diagonal. 

This is clearly a matrix eigenvalue problem the eigenvalues determine tJie vibrational frequencies and the eigenvectors are the normal modes of vibration. Typical output is shown in Figure 14.10, with the mass-weighted normal coordinates expressed as Unear combinations of mass-weighted Cartesian displacements making up the bottom six Unes. 

The latter equation is now in a standard form for determining the eigenvalues of the F matrix. The eigenvectors contained in C can then be backtransformed to the original coordinate system (C = S C )-... 

Thus far we have considered the eigenvalue decomposition of a symmetric matrix which is of full rank, i.e. which is positive definite. In the more general case of a symmetric positive semi-definite pxp matrix A we will obtain r positive eigenvalues where r general case we obtain a pxr matrix of eigenvectors V such that ... 

The PLS algorithm is relatively fast because it only involves simple matrix multiplications. Eigenvalue/eigenvector analysis or matrix inversions are not needed. The determination of how many factors to take is a major decision. Just as for the other methods the right number of components can be determined by assessing the predictive ability of models of increasing dimensionality. This is more fully discussed in Section 36.5 on validation. 

The eigenvalue-eigenvector decomposition of a Hermitian matrix with the complete orthonormal set of eigenvectors Vi and eigenvalues A, is written as... 

As indicated in Table 4.2, the eigenvalues of the Hessian matrix of fix) indicate the shape of a function. For a positive-definite symmetric matrix, the eigenvectors (refer to Appendix A) form an orthonormal set. For example, in two dimensions, if the eigenvectors are Vj and v2, v[v2 =0 (the eigenvectors are perpendicular to each other). The eigenvectors also correspond to the directions of the principal axes of the contours of fix). 

A. 1 Definitions / A.2 Basic Matrix Operations / A.3 Linear Independence and Row Operations / A.4 Solution of Linear Equations / A. 5 Eigenvalues, Eigenvectors / References /... 

The eigenvalue/eigenvector decomposition of the covariance matrix thus allows us to redefine the problem in terms of Nc independent, standard normal random variables 0in. 

In summary, given the covariance matrix C, an eigenvalue/eigenvector decomposition can be carried out to find U and A. These matrices define a linear transformation... 

Rotation of the translated factor axes is an eigenvalue-eigenvector problem, the complete discussion of which is beyond the scope of this presentation. It may be shown that there exists a set of rotated factor axes such that the off-diagonal terms of the resulting S matrix are equal to zero (the indicates rotation) that is, in the translated and rotated coordinate system, there are no interaction terms. The relationship between the rotated coordinate system and the translated coordinate system centered at the stationary point is given by... 

The Karhuhnen-LoSve transformation represents an eigenvalue-eigenvector calculation based upon the variance- covariance matrix of the features. It aims for a linear combination of features such that there are as much linear combinations as features. The linear combinations are mutually orthogonal and have a norm equal to one. Each linear combination (eigenvector) accounts for a part of the... 

IV. Eigenvalues and Eigenvectors of a Square Matrix An eigenvector of a matrix, M, is a vector such that... 

As in all matrix eigenvalue problems, we are able to express (n-1) elements of the eigenvector v (k) in terms of one remaining element. However, we can never solve for this one last element. So, for convenience, we impose one more constraint (equation to be... 

Solution of eigenvalue-eigenvector problems, where we find the eigenvalue x and the eigenvector u of the square symmetric matrix A such that... 

Principal component analysis is based on the eigenvalue-eigenvector decomposition of the n h empirical covariance matrix Cy = X X (ref. 22-24). The eigenvalues are denoted by > 2 — Vi > where the last inequality follows from the presence of same random error in the data. Using the eigenvectors u, U2,. . ., un, define the new variables... 

Overparameterization and frequently its sources are revealed by an eigenvalue-eigenvector analysis. In the module 1445 the matrix JT( )WJ(jB) is investigated. We call it normalized cross product matrix, because the partial derivatives are computed with respect to the normalized parameters... 

To investigate cases (i) and (ii), the S matrix obtained in Example 5.3 is used directly. Forming S S and applying eigenvalue-eigenvector decomposition (by the module M18), we obtain the results shown in Table 5.5. 

To investigate cases (iii) and (iv), we include only every second row of matrix S obtained in Example 5.3 when forming STS. Applying eigenvalue-eigenvector decomposition again, the results shown in Table 5.6 are obtained. 

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