Orthogonality of vectors

Orthogonalization of Vectors and its Relation to Cognitive Phenomena 3. A GEOMETRICAL VIEW... 

Another approach involves starting with an initial wavefimction Iq, represented on a grid, then generating // /q, and consider that tiiis, after orthogonalization to Jq, defines a new state vector. Successive applications //can now be used to define an orthogonal set of vectors which defines as a Krylov space via the iteration (n = 0,.. ., A)... 

Orthogonal transformations preserve the lengths of vectors. If the same orthogonal transformation is applied to two vectors, the angle between them is preserved as well. Because of these restrictions, we can think of orthogonal transfomiations as rotations in a plane (although the formal definition is a little more complicated). 

A set of complete orthonormal functions ipfx) of a single variable x may be regarded as the basis vectors of a linear vector space of either finite or infinite dimensions, depending on whether the complete set contains a finite or infinite number of members. The situation is analogous to three-dimensional cartesian space formed by three orthogonal unit vectors. In quantum mechanics we usually (see Section 7.2 for an exception) encounter complete sets with an infinite number of members and, therefore, are usually concerned with linear vector spaces of infinite dimensionality. Such a linear vector space is called a Hilbert space. The functions ffx) used as the basis vectors may constitute a discrete set or a continuous set. While a vector space composed of a discrete set of basis vectors is easier to visualize (even if the space is of infinite dimensionality) than one composed of a continuous set, there is no mathematical reason to exclude continuous basis vectors from the concept of Hilbert space. In Dirac notation, the basis vectors in Hilbert space are called ket vectors or just kets and are represented by the symbol tpi) or sometimes simply by /). These ket vectors determine a ket space. 

Any three-dimensional orthogonal coordinate system may be specified in terms of the three coordinates q, q2 and q3. Because of the orthogonality of the coordinate surfaces, it is possible to set up, at any point, an orthogonal set of three unit vectors ex, e2, e3, in the directions of increasing qx, q2, q3, respectively. It is important to select the qt such that the unit vectors define a right-handed system of axes. The set of three unit vectors defines a Cartesian coordinate system that coincides with the curvilinear system in... 

Because of its oscillatory component wave motion requires a related, but more complicated description than linear motion. The methods of particle mechanics use vectors to describe displacements, velocities and other quantities of motion in terms of orthogonal unit vectors, e.g. 

In the theory of optics this phenomenon is accounted for in terms of geometrical construction, but the physical picture is less convincing. Double refraction is a well-documented property of most crystals, at its most spectacular in Iceland spar. The double image of an object viewed through the crystal indicates the existence of two independent rays and not the components of a single ray. In mathematical terms the two rays are linearly independent and therefore orthogonal. Any intermediate situation represents a linear combination of the two orthogonal basis vectors and can be resolved into two components. What happens to an individual photon is however, not clear. 

The purpose of this Chapter is not to present an exhaustive theory of linear algebra that would take more than a volume by itself to be presented adequately. It is rather to introduce some fundamental aspects of vectors, matrices and orthogonal functions together with the most common difficulties that the reader most probably has encountered in scientific readings, and to provide some simple definitions and examples with geochemical connotations. Many excellent textbooks exist which can complement this introductory chapter, in particular that of Strang (1976). 

The space of the column-vectors x such as Ax=Om< where 0m is a m x m matrix of zeroes, is called the nullspace of the matrix A. Any vector from the nullspace is therefore orthogonal to any vector from the row-space. The left nullspace of A is the set of vectors ym such as yTA = 0 . Any vector from the left nullspace is therefore orthogonal to any vector from the column-space. The left nullspace of A is identical to the nullspace of AT. 

Each vector x can be decomposed as the sum of a vector from the row-space and a vector in the nullspace. These two vectors are orthogonal. Each vector ym can be decomposed as the sum of a vector from the column-space and a vector in the left nullspace. These two vectors are orthogonal. 

We now return to the very basic concept of vector length in vector spaces. In a Euclidian space spanned by a base of n orthogonal unit vectors eb the squared length l2 of a n-vector t> is the quadratic form given by... 

Now the expression Normal Equations starts to make sense. The residual vector r is normal to the grey plane and thus normal to both vectors f ,i and f , 2 As outlined earlier, in Chapter Orthogonal and Orthonormal Matrices (p.25), for orthogonal (normal) vectors the scalar product is zero. Thus, the scalar product between each column of F and vector r is zero. The system of equations corresponding to this statement is ... 

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. 

The residuals Ru are defined by the projections of the vectors C into the space U they are orthogonal to U. This projection is simpler due to the orthogonal base vectors. Ru is the closest the vectors C get to U. 

LDA uses a space that is defined by a basis set of vectors, called linear discriminants (LDs) that are similar to the PCs obtained from PCA analysis. Like PCs, LDs are linear combinations of the original M variables in the jc data that are also orthogonal to one another. However, they are determined using a quite different criterion where the ratio of between-class variability and within-class variability in the calibration data is maximized. 

By using the chain rule for differentiation and the orthogonality of the basis vectors, we then obtain... 

The mutual orthogonality of the character vectors is reminiscent of the axes of a Cartesian coordinate system, and suggests the valuable idea that the character vectors of a group form a basis for the symmetry. Any vector can be resolved into components of different symmetry types. The projection of any vector onto any symmetry species is calculable. So we have returned to the geometrical point of view ... 

Using the orthogonality of characters taken as vectors we can reduce the above set of characters to Ai + E. Hence, we say that our orbital set of three Ish orbitals forms a reducible representation consisting of the sum of Ai and E IR s. This means that the three Ish orbitals can be combined to yield one orbital of Ai symmetry and a pair that transform according to the E representation. 

Eq.(21) requires that all occupied ground state orbitals be orthogonal to a linear combination of the excited state orbitals b j lvij ), which describes an excited electronic state. Eq.(22) requires the orthogonality of all occupied excited state orbital associated with a spin functions to the arbitrary vector Y7 IVoi ) from the subspace of the occupied ground state orbitals associated with a spin functions. In general, the coefficients 6° can be determined by minimizing the excited state Hartree-Eock energy. However, calculations show that the choice... 

The orthogonality of all the vector spherical harmonics, which was established in the preceding section, implies that the coefficients in the expansion (4.23) are of the form... 

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