Eigenvectors of matrix

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. 

For slow exchange, a convenient matrix of eigenvectors is given by equation (B2.4.23). 

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 ... 

For data with many variables and a small number of objects (n is much smaller than in), the above SVD decomposition is very time consuming. Therefore, a more efficient algorithm avoids the computation of the eigenvectors of the m x m matrix X1 X. It can be shown that To is the matrix of eigenvectors of X X1, and P is the matrix of eigenvectors of XT X, both normalized to length 1 (see also Appendix A.2.7). 

After the number of factors have been determined, it is necessary to interpret the factors as physically real sources. For the applications of this approach to aerosol source identification (4,6-10), the reduced size matrix of eigenvectors was rotated in such a way as to maximize the number of values that are zero or unity. This rotation criterion, called "simple structure" is described in the appendix of reference 4. 

Suppose we want to carry out a change of variables that will eliminate all cross products (those with i k) from /, so as to have / expressed as a sum of squares. Let C be the square matrix of eigenvectors of A. We previously used X to denote this matrix, but to avoid confusion with the... 

As a check, we verify that a similarity transformation with the unitary square matrix of eigenvectors diagonalizes Sx ... 

With PCA, it is possible to build an empirical mathematical model of the data as described by Equation 4.4 where Tk is the n x k matrix of principal component scores and k is the m x A matrix of eigenvectors. 

For this alternative formulation, we define the matrix of eigenvectors b according to Equation 4.17. The corresponding principal component model is given by Equation 4.18. 

So if we consider the value of orthonormal polynomials P (E) at the energies Eg at which the first neglected polynomial vanishes, a new orthogonality relation is achieved from the orthogonality property of the matrix of eigenvectors ... 

PP is the unit matrix since P is an orthogonal matrix of eigenvectors and P = P This gives the following important relation which shows that any matrix can be factorized into two different matrices. 

Table 6.19 Decomposition of the cross —product matrix (Table 6.18) provides a matrix of eigenvectors and a vector of eigenvalues...

Fig. 6-15 Principal component analysis of multidimensional, chemical-genetic data, (a) Eigenvalues and associated variance, and eigenvectors and associated factor scores computed from the data in Fig. 6-14(a). The matrix of eigenvectors...

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