Orthogonalization vectors

Figure 5. For = 3, the vectors g, and h. Nascent (right-hand column) and orthogonalized (left-hand column) results at R (2.53). For orthogonal vectors = 0.0430, g = 0.0825, and h = 0,000233. Vectors are scaled for visual clarity.

Figure 6. For = 2, the orthogonal vectors g, h and seam vectors w, = 1-3. Orthogonal g and h and three seam coordinates at...

The matrix A in Eq. (7-21) is comprised of orthogonal vectors. Orthogonal vectors have a dot product of zero. The mutually perpendicular (and independent) Cartesian coordinates of 3-space are orthogonal. An orthogonal n x n such as matr ix A may be thought of as n columns of n-element vectors that are mutually perpendicular in an n-dimensional vector space. 

Two orthogonal vectors are orthonormal when, in addition to orthogonality, the norms of these vectors are equal to one ... 

Generally, two vectors that are orthogonal in S will be oblique in 5 , unless the vectors are parallel to the coordinate axes. This is illustrated in Fig. 32.4. Furthermore, if X and y are orthogonal vectors in S, then the vectors W x and W" y are orthogonal in 5 ,. This follows from the definition of orthogonality in the metric W (eq. 32.12) ... 

A matrix of order l has l2 elements. Each irreducible representation T, must therefore contribute If -dimensional vectors. The orthogonality theorem requires that the total set of Y f vectors must be mutually orthogonal. Since there can be no more than g orthogonal vectors in -dimensional space, the sum Y i cannot exceed g. For a complete set (19) is implied. Since the character of an identity matrix always equals the order of the representation it further follows that... 

What is the unperturbed Hamiltonian (H//(0)) whose eigenvectors are xa and Xb an(l whose eigenvalues are ea and eb We can construct this operator explicitly from the set of /U-orthogonal vectors xa and xb ... 

A square matrix O is orthogonal if its column-vectors form a set of orthogonal vectors and have unit length, i.e.,... 

Given a set of independent vectors x1 x2,. .., xn, it is requested to form a new set yk,y2, y , of orthogonal vectors spanning the same space. The Gram-Schmidt orthogonalization scheme sets... 

This procedure is important for producing orthogonal vectors and functions in numerical analysis. It is not unique since it can be started from any xk. 

The determination of the projections can be regarded as a linear least-squares fit only now we have an orthogonal set of vectors V =, as in Figure 5-28, rather than a general set of non-orthogonal vectors in F in the equivalent Figure 4-12. The projected test vector tproj is a linear combination of the vectors V. 

Now, the maximum number of dimensional vectors which can be orthogonal is g. Consider, for example, a two-dimensional vector space containing the two orthogonal vectors ... 

Since many of the basic arguments in Section 4.3 will lean heavily on the concept of orthogonal vectors in generalized, multidimensional space, a brief summary of the essentials will be included here. 

Direct use of this equation is not advisable, however, since it is only slowly convergent. A more efficient method is obtained by constructing a set of orthogonal vectors x as... 

In how many ways can these vectors be chosen We may choose the character xf k) from any of the Nr IRs. Therefore the number of mutually orthogonal vectors is the number of IRs, Nr and this must be < Nc the dimension of the space. In fact, we shall see shortly that the number of IRs is equal to the number of classes. 

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