Effective eigenvectors

We remark that, as opposed to Eqs. (145)—(129), in this construction two unitary transformations are needed to obtain the effective eigenvectors to second order (after the first transformation, where keeping only the diagonal blocks in Eq. (164) yields the eigenvalues to second order, but not the eigenvectors). 

This leads to an eigenvalue equation which is formally identical to (5.109) but with effective frequencies i.(tf), effective eigenvectors e(S) and effec-... 

When we regard each of our spectra as a unique point in the n-dimensional absorbance space, we can say that the error in our data is isotropic. By this, we mean that the net effect of the errors in a given spectrum is to displace that spectrum some random distance in some random direction in the n-dimensional data space. As a result, when we find the eigenvectors for our data, each eigenvector will span its equivalent share of the error. But recall, we said that we must take degrees-of-ffeedom into account in order to understand what is meant by equivalent share. 

The projection of the X data vectors onto the first eigenvector produces the first latent variable or pseudomeasurement set, Zx. Of all possible directions, this eigenvector explains the greatest amount of variation in X. The second eigenvector explains the largest amount of variability after removal of the first effect, and so forth. The pseudomeasurements are called the scores, Z, and are computed as the inner products of the true measurements with the matrix of loadings, a ... 

Owing to Eq. (36), the eigenvectors of the two effective Hamiltonians are connected by the transformation... 

Note that because the effective Hamiltonian matrix is not Hermitian, the eigenvectors are not orthogonal. However, when ac is small, the orthogonality properties are satisfactorily verified. 

Potential fluid dynamics, molecular systems, modulus-phase formalism, quantum mechanics and, 265—266 Pragmatic models, Renner-Teller effect, triatomic molecules, 618-621 Probability densities, permutational symmetry, dynamic Jahn-Teller and geometric phase effects, 705-711 Projection operators, geometric phase theory, eigenvector evolution, 16-17 Projective Hilbert space, Berry s phase, 209-210... 

The Matlab function PCR ca librat ion. m performs the PCR calibration according to equation (5.62). Note that we use ne=12 eigenvectors in the above calculations. This is the optimal number for prediction, as we show in Cross Validation (p.303j. The reader is invited to play with this number and observe the effect. 

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