Row space

For a given structure, the values of S at which in-phase scattering occurs can be plotted these values make up the reciprocal lattice. The separation of the diffraction maxima is inversely proportional to the separation of the scatterers. In one dimension, the reciprocal lattice is a series of planes, perpendicular to the line of scatterers, spaced 2Jl/ apart. In two dimensions, the lattice is a 2D array of infinite rods perpendicular to the 2D plane. The rod spacings are equal to 2Jl/(atomic row spacings). In three dimensions, the lattice is a 3D lattice of points whose separation is inversely related to the separation of crystal planes. 

RHEED is a powerful tool for studying the surface structure of crystalline samples in vacuum. Information on the surface symmetry, atomic-row spacing, and evidence of surfece roughness are contained in the RHEED pattern. The appearance of the RHEED pattern can be understood qualitatively using simple kinematic scattering theory. When used in concert with MBE, a great deal of information on film growth can be obtained. 

In the illustration of Fig. 29.4 we regard the matrix X as either built up from n horizontal rows of dimension p, or as built up from p vertical columns x,.of dimension n. This exemplifies the duality of the interpretation of a matrix [9]. From a geometrical point of view, and according to the concept of duality, we can interpret a matrix with n rows and p columns either as a pattern of n points in a p-dimensional space, or as a pattern of p points in an n-dimensional space. The former defines a row-pattern P" in column-space 5, while the latter defines a column-pattern P in row-space S". The two patterns and spaces are called dual (or conjugate). The term dual space also possesses a specific meaning in another... 

Fig. 29.5. Geometrical interpretation of an nxp matrix X as either a row-pattern of n points P" in p-dimensional column-space S (left panel) or as a column-pattern of p points / in n-dimensional row-space S" (right panel). The p vectors Uy form a basis of 5 and the n vectors v, form a basis of 5".

In the same Fig. 29.5, the row-space S" is shown as an -dimensional coordinate space in which each column Xj of X defines a point with coordinates Xy,.., x j)... 

Dimensions and rank of a matrix are distinct concepts. A matrix can have relatively large dimensions say 100x50, but its rank can be small in comparison with its dimensions. This point can be made more clearly in geometrical terms. In a 100-dimensional row-space S ° , it is possible to represent the 50 columns of the matrix as 50 points, the coordinates of which are defined by the 100 elements in each of them. These 50 points form a pattern which we represent by P °. It is clear... 

Fig. 29.7. Illustration of a pattern of points with rank of 2. The pattern is represented by a matrix X with dimensions 5x4 and a linear dependence between the three columns of X is assumed. The rank is shown to be the smallest number of dimensions required to represent the pattern in column-space 5 and in row-space S".

Fig. 29.8. (a) Pattern of points in column-space S (left panel) and in row-space S" (right panel) before column-centering, (b) After column-centering, the pattern in 5 is translated such that the centroid coincides with the origin of space. Distances between points in S are conserved while those in S" are not. (c) After column-standardization, distances between points in S and 5" are changed. Points in 5" are located on a (hyper)sphere centered around the origin of space. 

We have seen above that the r columns of U represent r orthonormal vectors in row-space 5". Hence, the r columns of U can be regarded as a basis of an r-dimensional subspace 5 of 5". Similarly, the r columns of V can be regarded as a basis of an r-dimensional subspace S of column-space 5. We will refer to S as the factor space which is embedded in the dual spaces S" and SP. Note that r

factor-spaces will be more fully developed in the next section. 

Similarly, Fig. 31.2b shows the column-pattern F of the p columns of the data table X by means of an elliptical envelope in the dual n-dimensional row-space 5". The ellipses should be interpreted as (hyper)ellipsoidal equiprobability envelopes of multinormal data. In practice the data are rarely multinormal and the centroid (or center of mass) of the pattern does not generally appear at the origin of space. An essential feature is that the equiprobability envelopes are similarly shaped in Figs. 31.2a and b. The reason for this will become apparent below. Note that in the previous section we have assumed by convention that n exceeds p, but this is not reflected in Figs. 31.2a and b. 

Fig. 31.2. Geometrical example of the duality of data space and the concept of a common factor space, (a) Representation of n rows (circles) of a data table X in a space Sf spanned by p columns. The pattern P" is shown in the form of an equiprobabi lity ellipse. The latent vectors V define the orientations of the principal axes of inertia of the row-pattern, (b) Representation of p columns (squares) of a data table X in a space y spanned by n rows. The pattern / is shown in the form of an equiprobability ellipse. The latent vectors U define the orientations of the principal axes of inertia of the column-pattern, (c) Result of rotation of the original column-space S toward the factor-space S spanned by r latent vectors. The original data table X is transformed into the score matrix S and the geometric representation is called a score plot, (d) Result of rotation of the original row-space S toward the factor-space S spanned by r latent vectors. The original data table X is transformed into the loading table L and the geometric representation is referred to as a loading plot, (e) Superposition of the score and loading plot into a biplot.

The same geometrical considerations can be applied to the dual representation of the column-pattern in row-space S" (Fig. 31.2b). Here u, is the major axis of symmetry of the equiprobability envelope. The projection of theyth column Xy of X upon u, is at a distance from the origin given by ... 

The vector of column-means nip defines the coordinates of the centroid (or center of mass) of the row-pattern P" that represents the rows in column-space Sf . Similarly, the vector of row-means m defines the coordinates of the center of mass of the column-pattern that represents the columns in row-space S". If the column-means are zero, then the centroid will coincide with the origin of SP and the data are said to be column-centered. If both row- and column-means are zero then the centroids are coincident with the origin of both 5" and S . In this case, the data are double-centered (i.e. centered with respect to both rows and columns). In this chapter we assume that all points possess unit mass (or weight), although one can extend the definitions to variable masses as is explained in Chapter 32. 

In Section 29.3 it has been shown that a matrix generates two dual spaces a row-space S" in which the p columns of the matrix are represented as a pattern P , and a column-space S in which the n rows are represented as a pattern P". Separate weighted metrics for row-space and column-space can be defined by the corresponding metric matrices and W. This results into the complementary weighted spaces and S, each of which can be represented by stretched coordinate axes using the stretching factors in -J v and, where the vectors w and Wp contain the main diagonal elements of W and W. ... 

Any data matrix can be considered in two spaces the column or variable space (here, wavelength space) in which a row (here, spectrum) is a vector in the multidimensional space defined by the column variables (here, wavelengths), and the row space (here, retention time space) in which a column (here, chromatogram) is a vector in the multidimensional space defined by the row variables (here, elution times). This duality of the multivariate spaces has been discussed in more detail in Chapter 29. Depending on the chosen space, the PCs of the data matrix... 

The pure variable technique can be applied in the column space (wavelength) as well as in the row space (time). When applied in the column space, a pure column is one of the column factors. In LC-DAD this is the elution profile of the compound which contains that selective wavelength in its spectrum. When applied in the row space, a pure row is a pure spectrum measured in a zone where only one compound elutes. 

L oxygen exposure with a (2 x 1) structure present image (b) is after 42 L oxygen exposure with both (2 x 1) and (3 x 1) states present line profiles of the rows running in the < 100 > direction also shown, inter-row spacings are twice and three times the Cu-Cu distance in the < 110 > direction (c). Also shown is the image of a c(6 x 2) structure present as a minor component (b, d). (Reproduced from Ref. 16). 

Now clearly the rows of K are linear combinations of rows of M, and since they are obviously linearly independent, the row spaces of K and M must be the same. Hence, from Eq. (12) we have... 

Figure 14-3 (a) The representation of two columns of a matrix in row space. The vector sum of the two column vectors is the first principal component (PCI), (b) A close-up view of Figure 14-3a, illustrating the line segments, direction angles, and projection of Columns 1 and 2 onto the first principal component. 

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. 

Corundum-type Magnetic Oxide Surfaces. The substrate hematite with the corundum-type crystal structure is an antiferromagnet below 963 K. In the corundum-type structure of hematite, pairs of ferric ions are in a row spaced by single vacant sites along the <111> direction. The positions of ferric ions in each pair are shifted slightly upward or downward in the <111> direction. We denote these lattice positions as up and down sites (Au and A ), respectively. 

The field yield of a crop is determined by the yield of each plant, as well as each plant s density or stand (plants per unit area). Optimisation of the row spacing can increase the stand without negatively affecting the plant yield, hence increasing the field yield (Keren et al. 1983). 

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