Normalized eigenvectors

Thus, if we wish to compare the eigenvectors to one another, we can divide each one by equation [57] to normalize them. Malinowski named these normalized eigenvectors reduced eigenvectors, or REV". Figure 52 also contains a plot of the REV" for this isotropic data. We can see that they are all roughly equal to one another. If there had been actual information present along with the noise, the information content could not, itself, be isotropically distributed. (If the information were isotropically distributed, it would be, by definition, noise.) Thus, the information would be preferentially captured by the earliest... 

Here the symbol q, indicates the 3-dimensional vector position of the f particle. To apply Hilbert space concepts to this theory we now postulate a one-to-one correspondence between the rays of and the points of the Schrodinger 3N hyperspace. Thus there exists a continuum of normalized eigenvectors in represented by the symbol 4i.4a.". > ... 

If the normalized eigenvector n) is known, these relations may be used to obtain the eigenvectors — 1) and n -h 1), both of which will be normalized. 

The q s are the coordinates along the normalized eigenvectors, in the initial state and in the saddle point, respectively, so that x = Y. . The /s are force constants... 

An important application of eigenanalysis is the diagonalization of a (symmetric) matrix A. Let U denote the matrix whose columns are the normalized eigenvectors > 2<- - By the definition (1.60) we have... 

Since (3.18) has the same form as (3. 1), we can apply the results obtained above to each eigenvalue of B0) in place of the eigenvalue do of H0 hitherto considered. By hypothesis B y) has to distinct eigenvalues k, . .., Xm when considered in the subspace Eofy. Let reduced resolvent of Bm belonging to Xj (considered in the subspace Eq X) ... 

Next we turn to the estimation of the eigenvectors. Let associated with the eigenvalues Zj% (j— 1,. .., m). phase factor since are different from each other by (10. 7). Noting that (R% — Xv. ) wc have... 

The normalized eigenvectors / are the canonical variables for the matrix X similar to the factor loadings in PCA. 

Let U denote the matrix of normalized eigenvectors Uy of S S such that U U = I. The new set of parameters a are called principal components. The objective function can also be expressed in terms of the principal components. 

Any real square matrix has N (complex) eigenvalues (not necessarily different), and at most N linearly independent eigenvectors. It can be proved that all eigenvalues of a symmetric matrix A are real numbers, and A has exactly N different eigenvectors. The normalized eigenvectors of a symmetric matrix form an orthonormal set of vectors... 

In other words, the normalized eigenvectors of a symmetric matrix form an orthonormal basis in Euclidean space Ej. ... 

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