Schmidt orthogonalization process

If we choose our basis functions for a particular function space to be orthonormal (orthogonal and normalized) i.e. (/ /,) = J/, /, dr then, since the transformation operators are unitary ( 5-7), the representation created will consist of unitary matrices. This is proved in Appendix A.6-1. It should be stated that it is always possible to find an orthonormal basis and one way, the Schmidt orthogonalization process, is given in Appendix A.6-2. 

Gram - Schmidt orthogonalization process Thus, equation (B.6) takes the form... 

It is evident from these discussions that population balance equations are important in the description of dispersed-phase systems. However, they are still of limited use because of difficulties in obtaining solutions. In addition to the numerical approaches, solution of the scalar problem has been via the generation of moment equations directly from the population balance equation (H2, H17, R6, S23, S24). This approach has limitations. Ramkrishna and co-workers (H2, R2, R6) presented solutions of the population balance equation using the method of weighted residuals. Trial functions used were problem-specific polynomials generated by the Gram-Schmidt orthogonalization process. Their approach shows promise for future applications. 

Scattering processes, 586 Schiff, L. J., 437,444 Schiffer, M.% 363 SchUchting, H., 24 Schmidt orthogonalization, 65 Schonflies notation for magnetic point groups, 739... 

The simple Gram-Schmidt ON-process could have been applied to any initial basis set to find a set of solutions to (3.1.15). The equations above are all equivalent. Any two solutions to these equations are related by some orthogonal matrix V ... 

The Gram-Schmidt orthogonalization of the frequency independent vectors (j), A(j) produces the orthonormal basis qi, , qm by the Lanczos process so that... 

J"his can be done by a standard method known as the Schmidt orthogonal-ization process for vectors. The new vectors are defined by equatioii.s... 

Orthogonality theorem for irreducible representations, 341/. Orthogonalization process, Schmidt, 212 Out-of-plane bending, 58 Overtone frequencies, 9, 36, 246 selection ndes, infrared, 160 Uaman, 161 Overtone levels, 36... 

The new function needs to be renormalized.) This process, known as Schmidt orthogonalization, may be generalized and applied sequentially to any number of linearly independent functions. 

The result is three (not two ) different wavefunctions, which apparently describe a doubly degenerate situation. The problem here is that 0o(e), 0 (e) and are not orthogonal. The solution is to use a technique known as the Schmidt orthogonal-ization process. First we accept equation 4.43 as one component of the doubly degenerate pair. 

Nonorthogonality complicates the matrix diagonalization process see Of-fenhartz, pp. 338-341 for the procedure used. (Note that we can, if we like, use the Schmidt or some other orthogonalization procedure to form orthogonal linear combinations of the nonorthogonal basis functions and then use these orthogonalized basis functions to form the secular equation.)... 

The process of orthgonalization is somewhat analogous to the orthgonalization of vectors, known as the Gram-Schmidt procedure. Construction of orthogonal descriptors 2, 2, 2,... from given descriptors D, D, D,... is accomplished by calculating the residuals between the descriptors. " By definition, the residual of a correlation between two descriptors is the part of one descriptor that does not correlate with the other. 

Show that any real positive-definite matrix M can be written as M = U U, udiere U is real and upper-triangular, thus formalizing the Schmidt process (p. 33). %ow also that in this case the calculation of the inverse matrix, needed in relating orthogonal and non-orthogonal bases, is particularly simple. 

To do this, we first note an important simplication in any metric space, the Schmidt process (p. 33) can be used to ensure that two subspaces (in this case t, and %) are orthogonal. We therefore assume that (by adding suitable multiples of the i i operators) the new % operators yield scalar-product matrices i i % and i 3 i i that are identically zero. This means, since i)i = (a al aJ. ..), that a, P = aj = 0 for... 

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