Row vectors in column space

Figure 14-2 The representation of column vectors in row space of matrix M...

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. 

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. 

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 determinant can be regarded as a measure of the volume which is spanned by the column vectors (or row vectors) of the matrix in the vector space. For example, in a two dimensional space, two vectors can span a surface area, provided that the vectors are not parallel. If they should be parallel, the surface area would be zero. In the three-dimensional space, three vectors can span a volume, provided that they do not lie in the same plane. If they should do so, the volume would be zero. In that case, any of the three vectors can be expressed as a linear combination of the two others the vectors are said to be linear dependent. 

The quantities that are entries in the two-dimensional list are called matrix elements. The brackets written on the left and right are included to show where the matrix starts and stops. If m — n, we say that the matrix is a square matrix. A vector in ordinary space can be represented as a list of three components. We consider a matrix with one row and n columns to be a row vector. We consider a matrix with m rows and one column to be a column vector. We now refer to ordinary numbers as scalars, to distinguish them from vectors and matrices. 

A metric tensor with matrix 9pq is obviously symmetrical and regular (this last assertion is necessary and sufficient for the linear independence of gp in the basis of k orthonormal vectors in this space, we obtain det g , as a product of two determinants first of them having the rows and second one having the columns formed from Cartesian components of gp and gq. Because of the linear independence of these k vectors, every determinant and therefore also det g , is nonzero and conversely). Contravariant components gP of the metric tensor are defined by inversion... 

B.2.1.8 Row and Column Space The column space of a matrix A is the vector space generated by all linear combinations of the column vectors of A. Hence, the column space of A is equal to the span of the columns of A. Similarly, the row space of matrix A is the vector space that is generated by all combinations of the row vectors of A. The dimension of the column space is thus equal to the number of linearly independent column vectors in A, whereas the dimension of the row space of A is equal to the number of linearly independent row vectors in A, which are both equal to the rank of A, rank(A). Hence, the dimension of the column space of... 

Vectors in ordinary space have three components. However, row and column vectors can have a number of elements other than three, just as a matrix can have a number of rows and columns other than three if they represent something other than a vector in three-dimensional space. Matrix multiplication with fairly large matrices can involve a lot of computation. Computer programs can be written to carry out the process, and such programs are built into Mathematica and into computer languages such as BASIC so that a matrix multiplication can be carried out with a single statement. 

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".

Summarizing, we find that, depending on the choice of a and p, we are able to reconstruct different features of the data in factor-space by means of the latent vectors. On the one hand, if a = 1 then we can reproduce the cross-products C between the rows of the table. On the other hand, if p equals 1 then we are able to reproduce the cross-products between columns of the table. Clearly, we can have both a = 1 and P = 1 and reproduce cross-products between rows as well as between columns. In the following section we will explain that cross-products can be related to distances between the geometrical representations of the corresponding rows or columns. 

In Fig. 31.2a we have represented the ith row x, of the data table X as a point of the row-pattern F in column-space S . The additional axes v, and V2 correspond with the columns of V which are the column-latent vectors of X. They define the orientation of the latent vectors in column-space S. In the case of a symmetrical pattern such as in Fig. 31.2, one can interpret the latent vectors as the axes of symmetry or principal axes of the elliptic equiprobability envelopes. In the special case of multinormally distributed data, Vj and V2 appear as the major and minor... 

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.

Once we have obtained the projections S and L of X upon the latent vectors V and U, we can do away with the original data spaces S and 5". Since V and U are orthonormal vectors that span the space of latent vectors each row i and each column j of X is now represented as a point in as shown in Figs. 31.2c and d. The... 

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. ... 

From the latent vectors and singular values one can compute the nxr generalized score matrix S and the pxr generalized loading matrix L. These matrices contain the coordinates of the rows and columns in the space spanned by the latent vectors ... 

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... 

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. 

The matrix Y can be regarded as a row or as a column matrix and consequently we can also concentrate on the columns of Y rather than the rows. The columns are linear combinations of the concentration profiles of the species and they all lie in a 3-dimensional space as well. The columns of the matrix U form a basis in this space. And the coordinates of each column vector of Y are contained in the columns of the matrix SV. 

The columns (or rows) of a unitary matrix are related to a set of orthogonal normalized vectors in a general vector space. If, for example,... 

Note that the column vectors v, and the adjoint row vectors v J live in different spaces, so it makes no sense, for example, to add v, and vj. However, according to the rule (9.11) of matrix multiplication, it makes perfect sense to multiply an adjoint v row vector (with... 

In Section 1.2, we pointed out that an ordered set of n numbers can be regarded as defining a vector in an abstract /i-dimensional space. In line with this idea, a row matrix is often called a row vector, and a column matrix is often called a column vector. The general m by n matrix (2.1) can be regarded as either a set of m ordered row vectors or a set of n ordered column vectors. 

Let us consider a vector in ordinary three-dimensional space. We can specify the length and direction of this vector in the following way. We arrange to have one end of the vector lie at the origin of a Cartesian coordinate system. The other end is then at a point which may be specified by its three Cartesian coordinates, x, y, z. In fact, these three coordinates completely specify the vector itself provided it is understood that one end of the vector is at the origin of the coordinate system. We can then write these three coordinates as a column matrix, in this case one with three rows, x y z, and say that the matrix represents the vector in question. 

If a matrix contains only one row, it is called a row matrix or a row vector. The matrix B shown above is an example of a 1 X 3 row vector. Similarly, a matrix containing only one column is known as a column matrix or column vector. The matrix C shown above is a 6 X 1 column vector. One use of vectors is to represent the location of a point in an orthogonal coordinate system. For example, a particular point in a three-dimensional space can be represented by the 1X3 row vector... 

Let us postulate that we live in a 3D hypersurface that slides along the u axis with speed v°u = ca, where the u axis coincides with the arrow of time. The 4-velocity is then a (row or column) vector 1 a = ( ca,vx,vy,vz). The plus (resp. minus) sign corresponds to the speed of preons that enter (resp. leave) our 3D world, parallel (resp. antiparallel) to the time arrow. It will be seen below that this constant ca is the one that enters Einstein s mass-energy equation, and corresponds to the speed of our 3D world along the time axis (interpretation 2 in Fig. 1). The speed of electromagnetic radiation in free space is a different constant c. The value of the latter may be either identical or numerically close to c , depending of whether one adopts a relativistic or an emission theory for photons, respectively (see Section V). 

Equation (7) describes the transformation of the set of basis vectors ei e2 e31 that are firmly embedded in configuration space and were originally coincident with fixed orthonormal axes x y z prior to the application of the symmetry operator R(n (3 7). In eq. (8) the column matrix x y z) contains the variables x y z, which are the components of the vector r = OP and the coordinates of the point P. In eq. (9) the row matrix (x y z contains the functions x y z (for example, the angle-dependent factors in the three atomic p functions px, py, pz). 

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