Matrices notation

A useful first step towards the fitting of more complex linear functions, is to translate the equations into a matrix oriented notation. Equation (4.5) is actually a system of m equations, where m is the number of (x,y)-data pairs. 

This system of m equations can be written as one matrix equation. 

Similarly, the vector of residuals r, as introduced in equation (4.1), can be defined in a matrix equation  

The task is to find that vector a for which ssq is minimal. 

Now we are in a better position to generalise to more complex linear functions. The prototype of linear least-squares fitting is the fitting of a 

We can solve Eq. (13.8) also by the methods of linear algebra. We place the coefficients 

In matrix notation, a system of linear equations can be written down as 

The first row is the derivative with respect to carbon (9 /9C), the second row is the derivative with respect to oxygen (9 /90), the third row is the derivative with respect to hydrogen (9 /dH). Since there are three unknown coefficients, we need only the information from three equations, which are obtained by forming the derivatives. We could likewise instead of one derivatives take also the derivative with respect to potassium (9 /dK), and we would obtain the same final result. 

When we apply this method of matrix inversion, we must be sure that the procedure will work. This means that we have chosen the unknowns and the equations appropriate, finally there should be a nontrivial solution, i.e., the unknown variables X, y, z must be different from zero. 

We have renamed the coefficients, and we have added X4 as the stoichiometric coefficient for water. We treat now Xi, as a vector and the coefficients of the 

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Written in matrix notation, the system of first-order differential equations, (A3.4.139) takes the fomi... 

A completely difierent approach to scattering involves writing down an expression that can be used to obtain S directly from the wavefunction, and which is stationary with respect to small errors in die waveftmction. In this case one can obtain the scattering matrix element by variational theory. A recent review of this topic has been given by Miller [32]. There are many different expressions that give S as a ftmctional of the wavefunction and, therefore, there are many different variational theories. This section describes the Kohn variational theory, which has proven particularly useftil in many applications in chemical reaction dynamics. To keep the derivation as simple as possible, we restrict our consideration to potentials of die type plotted in figure A3.11.1(c) where the waveftmcfton vanishes in the limit of v -oo, and where the Smatrix is a scalar property so we can drop the matrix notation. 

The index J can label quantum states of the same or different chemical species. Equation (A3.13.20) corresponds to a generally stiff initial value problem [42, 43]. In matrix notation one may write ... 

A more general notation than Wood s is available for all kinds of unit eells, ineluding those that are sheared, so that the superlattiee unit eell ean take on any shape, size and orientation. It is the matrix notation, defined... 

In matrix notation PCA approximates the data matrix X, which has n objects and m variables, by two smaller matrices the scores matrix T (n objects and d variables) and the loadings matrix P (d objects and m variables), where X = TPT... 

For each IT), Equation (2.46) generates a corresponding equation and collectively these equations can be shown using matrix notation as... 

These are just the secular equations shown in equation set (7-2) with F in place of H and the stacked matrix Eq. (7-6) of eigenvectors in place of a single eigenvector. In matrix notation... 

To see how cahbration can be extended to multicomponent analysis, the linear model of equation 10 can be generalized to accommodate several analytes in the same sample, and several measurements made on each sample. Expressed in matrix notation, this becomes... 

In the analysis of composites it is convenient to use matrix notation because this simplifies the computations very considerably. Thus we may write the above equations as (see Appendix E)... 

Using matrix notation, equation (3.20) may be transposed to give the stresses as a function of the strains... 

Using matrix notation, these three steps are canied out as follows ... 

Multiplying from the left by a specific basis function and integrating yields the Roothaan-Hall equations (for a closed shell system). These are the Fock equations in the atomic orbital basis, and all the M equations may be collected in a matrix notation. 

Linear Programming.28—A linear programming problem as defined in matrix notation requires that a vector x 0 (non-negativity constraints) be found that satisfies the constraints Ax <, b, and maximizes the linear function cx. Here x = (xx, , xn), A = [aiy] (i = 1,- -,m j = 1,- , ), b - (61 - -,bm), and c = (cu- -,c ) is the cost vector. With the original (the primal) problem is associated the dual problem yA > c, y > 0, bij = minimum, where y yx,- , ym)-A duality theorem 29 asserts that if either the primal or the dual has a solution then the values of the objective functions of both problems at the optimum are the same. It is a relatively easy matter to obtain the solution vector of one problem from that of the other. 

Linear algebraic problem, 53 Linear displacement operator, 392 Linear manifolds in Hilbert space, 429 Linear momentum operator, 392 Linear operators in Hilbert space, 431 Linear programming, 252,261 diet problem, 294 dual problem, 304 evaluation of methods, 302 in matrix notation, simplex method, 292... 

The system of Eq. (87) can be written in a more general matrix notation ... 

Equation (1.25) leads to the rath dimensional equivalent of eq. (1.13), which can be written in matrix notation as... 

This procedure would generate the density amplitudes for each n, and the density operator would follow as a sum over all the states initially populated. This does not however assure that the terms in the density operator will be orthonormal, which can complicate the calculation of expectation values. Orthonormality can be imposed during calculations by working with a basis set of N states collected in the Nxl row matrix (f) which includes states evolved from the initially populated states and other states chosen to describe the amplitudes over time, all forming an orthonormal set. Then in a matrix notation, (f) = (f)T (t), where the coefficients T form IxN column matrices, with ones or zeros as their elements at the initial time. They are chosen so that the square NxN matrix T(f) = [T (f)] is unitary, to satisfy orthonormality over time. Replacing the trial functions in the TDVP one obtains coupled differential equations in time for the coefficient matrices. 

For reasons that will be apparent, Eq. 3a is cast in matrix notation as follows. 

Partial least squares regression (PLS). Partial least squares regression applies to the simultaneous analysis of two sets of variables on the same objects. It allows for the modeling of inter- and intra-block relationships from an X-block and Y-block of variables in terms of a lower-dimensional table of latent variables [4]. The main purpose of regression is to build a predictive model enabling the prediction of wanted characteristics (y) from measured spectra (X). In matrix notation we have the linear model with regression coefficients b ... 

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