Row vectors

The first equation (1) is the equation of state and the second equation (2) is derived from the measurement process. Finally, G5 (r,r ) is a row-vector that takes the three components of the anomalous ciurent density vector Je (r) = normal component of the induced magnetic field. This system is non hnear (bilinear) because the product of the two unknowns /(r) and E(r) is present. 

Vector and Matrix Norms To carry out error analysis for approximate and iterative methods for the solutions of linear systems, one needs notions for vec tors in iT and for matrices that are analogous to the notion of length of a geometric vector. Let R denote the set of all vec tors with n components, x = x, . . . , x ). In dealing with matrices it is convenient to treat vectors in R as columns, and so x = (x, , xj however, we shall here write them simply as row vectors. 

There is one term for each Auj in the row vector which is in the curly braces (]. These terms are caUed constrained derivatives, which tehs how the object func tion changes when the independent variables Uj are changed while keeping the constraints satisfied (by varying the dependent variables Xi). 

The superscript t denotes a transposition of the r-vector, i.e. converting it from a column to a row vector. The rr notation for the quadmpole moment therefore indicates a 3 x 3 matrix containing the products of the x-, y- and z-coordinates, e.g. the Qxy component is calculated as the expectation value of xy. 

A matrix is a rectangular array of numbers, its size being determined by the number of rows and columns in the array. In this context, the primary concern is with square matrices, and matrices of column dimension 1 (column vectors) and row dimension 1 (row vectors). 

Using row-wise organization, an absorbance matrix holds the spectral data. Each spectrum is placed into the absorbance matrix as a row vector ... 

Where A,w is the absorbance for sample s at the wlh wavelength. If we were to measure the spectra of 30 samples at 15 different wavelengths, each spectrum would be held in a row vector containing 15 absorbance values. These 30 row vectors would be assembled into an absorbance matrix which would be 30 X 15 in size (30 rows, 15 columns). 

Another way to visualize the data organization is to represent the row vector containing the absorbance spectrum as a line drawing —... 

Where C is the concentration for sample s of the c 11 component. Suppose we were measuring the concentrations of 4 components in each of the 30 samples, above. The concentrations for each sample would be held in a row vector... 

In general, because the noise in the concentration data is independent from the spectral noise, each optimum factor, W, will lie at some angle to the plane that contains the spectral data. But we can find the projection of each W, onto the plane containing the spectral data. These projections are called the spectral factors, or spectral loadings. They are usually assigned to the variable named P. Each spectral factor P, is usually organized as a row vector. 

This provides an inductive, and a constructive, proof of the possibility of a triangular factorization of the specified form, provided only certain submatrices are nonsingular. For suppose first, that Au is a scalar, A12 a row vector, and A21 a column vector, and let Ln = 1. Then i u = A1U B12 — A12, and L2l and A22 axe uniquely defined, provided only Au = 0. But Au can be made 0, at least after certain row permutations have been made. Hence the problem of factoring the matrix A of order n, has been reduced to the factorization of the matrix A22 of order n — 1. 

Still another interpretation can be made by taking A22 to be a scalar, hence A21 a row vector and A12 a column vector. Suppose A1X has been inverted or factored as before. Then L21, R12, and A22 are obtainable, the two triangular matrices are easily inverted, and their product is the inverse of the complete matrix A. This is the basis for the method of enlargement. The method is to start with aai which is easily inverted apply the formulas to... 

When the operators A and B in Eq. (2.7) are sin q)le creation and annihilation operators the resulting propagator is called electron (nopagator or one-particte Green s function, and = -t-1. Collecting all these creation and amiihi-lation operators in a row vector a, the electron propagator can be expressed as. 

In particular, and c,-,- represent the sums of squared elements of the row-vectors X, and X,-. Consequently, and d > are the norms (or lengths) of the /th and / -th rows, respectively. The relation that links the three expressions of eq. (31.31) together is called the triangle equation or cosine rule ... 

Let us now consider a new set of values measured for the various X-variables, collected in a supplementary row vector x. From this we want to derive a row vector y of expected T-values using the predictive PLS model. To do this, the same sequence of operations is followed transforming x into a set of factor scores r, t 2, t A pertaining to this new observation. From these t -scores y can be... 

MATLAB is most at home dealing with arrays, which we will refer to as matrices and vectors. They are all created by enclosing a set of numbers in brackets, [ ]. First, we define a row vector by entering in the MATLAB Command Window ... 

We can also generate the row vector with the colon operator ... 

The uncertainty in multivariate calibration is characterized with respect to the evaluation functions at Eqs. (6.75) and (6.79). The prediction of a row vector x of dimension n from a row vector y of dimension m results from... 

The mathematical development of estimating correlation is taken from Liu s work and is a simple treatment. In 2DLC, we can form a retention matrix composed of the scaled k values for each zone so that each dimension (i.e., each unique column) has a k row vector for the compounds that were separated on it. In two dimensions, the retention representation becomes a matrix, ft, so that k takes the form for a four-component mixture ... 

Let us begin by representing a row matrix M = (1,2,3) in column space as shown in Figure 14-1. Note that the row vector M = (1,2, 3) projects onto the plane defined by columns 1 and 2 as a point (1, 2) or a vector (straight line) with a Cx direction angle (a) equal to... 

Figure 14-1 A representation of a row vector M = [1,2, 3] in column space, and the projection of this vector onto the plane represented by Columns 1 and 2.

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