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Matrix algebra functions

In multivariate least squares analysis, the dependent variable is a function of two or more independent variables. Because matrices are so conveniently handled by computer and because the mathematical formalism is simpler, multivariate analysis will be developed as a topic in matrix algebra rather than conventional algebra. [Pg.80]

The summations in Eqs. (2-73) are over all i. Equations (2-73) are called the normal regression equations. With the experimental observations of 3, as a function of the Xij, the summations are carried out, and the resulting simultaneous equations are solved for the parameters. This is usually done by matrix algebra. Define these matrices ... [Pg.43]

Entropy and Equilibrium Ensembles.—If one can form an algebraic function of a linear operator L by means of a series of powers of L, then the eigenvalues of the operator so formed are the same algebraic function of the eigenvalues of L. Thus let us consider the operator IP, i.e., the statistical matrix, whose eigenvalues axe w ... [Pg.470]

This completes the derivation of the derivatives needed for the gradient of the energy functional. The compact matrix forms of these results can be manipulated using matrix algebra and are readily implemented using optimized computer subroutine libraries. [Pg.37]

Vector multiplication can be accomplished easily by the use of one of Excel s worksheet functions for matrix algebra, MMULT(inafr/x7, matrix2). For the matrices A and B defined above. [Pg.188]

The arguments used by Heisenberg in formulating his quantum mechanics are extremely interesting. We shall not present them, however, nor enter into an extensive discussion of matrix mechanics, but shall give in the following sections a brief treatment of matrices, matrix algebra, the relation of matrices to wave functions, and a few applications of matrix methods to quantum-mechanical problems. [Pg.417]

The FWT basis malri.x. The fast wavelet transform (FWT) can be formulated in terms of matrix algebra by storing each of the wavelet functions in the time/wavelength domain in a matrix B. This matrix contains all the translations and dilations of the wavelet necessary to perform a full transform. One common way to organise this matrix is to sort the sets of shifted basis... [Pg.375]

It is possible to relate brain chemistry and behavior by means of matrix algebra. One can relate specific neurotransmitters and hormones to specific mental functions in vertical axis (columns) and mental processes on the horizontal axis (rows). The entries in the matrix can be used to relate the relative roles of different complex systems of neurotransmitters to different mental functions. Matrices have long been used to solve multiple linear equations, and they have played an important role in the development of computers after World War II. [Pg.95]

An efficient way to treat such a system is to assemble all coefficients of the different terms of the mass-balance equations in a matrix and to apply methods of matrix algebra to solve the system for steady-state concentrations (level III) or for the concentrations as functions of time (level IV) [19]. We denote the matrix of coefficients (the fate matrix ) by S, the vector of concentrations in all boxes of the model by c, and the vector of all source terms by q. The set of mass-balance equations describing the temporal changes of the concentrations in all boxes then reads c = -S c + q. The steady-state solution is obtained by setting c equal to zero and solving for c. This leads to ss -1. j obtain the steady-state concentrations the emission vector has to be multiplied by the inverse of the matrix S. For the dynamic solutions of the system, the eigenvalues and eigenvectors of S have to be determined. [Pg.127]

The linear variation method is the most commonly used method to find approximate molecular wave functions, and matrix algebra gives the most computationally efficient method to solve the equations of the linear variation method. If the functions /i, in the linear variation function [Pg.228]

The procedure for using matrix algebra to solve the linear variation equations when nonorthonormal basis functions are used is outlined in Problem 8.49. [Pg.235]

One starts with guesses for the occupied-MO expressions as linear combinations of the basis functions, as in (13.156). This initial set of MOs is used to compute the Fock operator F from (13.149) to (13.152). The matrix elements (13.158) are computed, and the secular equation (13.159) is solved to give an initial set of c, s.These c, s are used to solve (13.157) for an improved set of coefficients, giving an improved set of MOs, which are then used to compute an improved F, and so on. One continues until no further improvement in MO coefficients and energies occurs from one cycle to the next. The calculations are done using a computer. (The most efficient way to solve the Roothaan equations is to use matrix-algebra methods see the last part of this section.)... [Pg.429]

Operator algebra shares the characteristic with matrix algebra indeed, matrices can be considered as representations of operators in a given set of functions (coordinate system). [Pg.531]


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See also in sourсe #XX -- [ Pg.323 ]

See also in sourсe #XX -- [ Pg.323 ]




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