Functions of matrices

In many practical situations we have to compute a function / (A) of an x TV matrix A. A popular way of computing a matrix function is through the truncated Taylor series approximation. The conditions under which a matrix function / (A) has a Taylor series representation are given by the following theorem (Golub and Van Loan, 1996). 

Eigenvectors Vi (i = 1,. N) form an orthonormal basis in Ejv- Therefore, vector c can be represented as 

Substituting (E.18) into (E.17) and taking into account expansion (E.16), we obtain  

the approximation problem has been reduced to discrete polynomial approximation of the function / with nodes A and weights cf. To solve this problem it is critical to find the eigenvalues of A. 

Given a Hermitian matrix A, we can define a function of A, i.e., /(A), in much the same way we define functions f(x) of a simple variable x. For example, the square root of a matrix A, which we denote by A, is simply that matrix which when multiplied by itself gives A, i.e., 

The sine or the exponential of a matrix are defined by the Taylor series of the function, e.g., 

After these definitions, we are still faced with the problem of calculating A or exp (A). If A is a diagonal matrix 

What do we do if A is not diagonal Since A is Heimitian, we can always find a unitary transformation that diagonalizes it, i.e., 

If the above procedure were to yield a result for /(A) that was infinite, then /(A) does not exist. For example, if we try to calculate the inverse of a matrix A that has a zero eigenvalue (say aj = 0), then /(a,) = 1/aj = oo and so A does not exist. As Exercise 1.12(a) shows, the determinant of a matrix is just the product of its eigenvalues. Thus if one of the eigenvalues of A is zero, det(A) is zero and the above argument shows that A does not exist. This same result was obtained in a different way in Exercise 1.7. 

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The multivariate autocorrelation function should contain the total variance of these autocorrelation matrices in dependence on the lag x. Principal components analysis (see Section 5.4) is one possibility of extracting the total variance from a correlation matrix. The total variance is equal to the sum of positive eigenvalues of the correlation matrices. This function of matrices is, therefore, reduced into a univariate function of multivariate relationships by the following instruction ... 

The most common and usually the fastest way to calculate the exponential function of matrices is done by renormalisation followed by Pade-approximation to fifth order and then successive multiplication.108 109... 

N. J. Higham, Functions of Matrices Theory and Computation. SIAM, Philaderphia, 2008. 

In many cases it is very difficult to compute functions of matrices from their definitions... 

Some functions of matrices are defined as expansions. If /(x) is a function of a scalar variable x that has the expansion... 

Such functions of general matrices must usually be approximated by truncating the expansion at some practicable order. Functions of matrices satisfying the relation [A, A ] = 0 may be evaluated exactly, however, by using the relation A = UAU to give... 

Many parameters in the modeling of soil physies exhibit nonlinear relationships. One sueh relationship is the water retention rate as a function of matric potential, which defines for water flow a term similar to electric potential for eurrent flow. (See J. Nat. Resour. Life Sci. Educ., Vol. 27, 1998, pp. 13-19 for a deseription of the data set for this example.) Figure 9.45 shows experimental data for soil volume water content vs. matric potential. The data points are approximately equally spaced on a log scale of the independent variable. From the data points plotted on a log scale it seems apparent that the data points should all be weighted equally and this is the default used in all the previous data fittings. This will occur naturally for the data fitting even when the points are non-uniformly spaced along the x-axis. 

For a multidimensional function, the variable x is replaced by the vector x and matrices are used for the various derivatives. Thus if the potential energy is a function of 3N... 

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. 

Theoretical results of similar quality have been obtained for thermodynamics and the structure of adsorbed fluid in matrices with m = M = 8, see Figs. 8 and 9, respectively. However, at a high matrix density = 0.273) we observe that the fluid structure, in spite of qualitatively similar behavior to simulations, is described inaccurately (Fig. 10(a)). On the other hand, the fluid-matrix correlations from the theory agree better with simulations in the case m = M = S (Fig. 10(b)). Very similar conclusions have been obtained in the case of matrices made of 16 hard sphere beads. As an example, we present the distribution functions from the theory and simulation in Fig. 11. It is worth mentioning that the fluid density obtained via GCMC simulations has been used as an input for all theoretical calculations. 

We conclude, from the results given above, that both the ROZ-PY and ROZ-HNC theories are sufficiently successful for the description of the pair distribution functions of fluid particles in different disordered matrices. It seems that at a low adsorbed density the PY closure is preferable, whereas... 

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 ... 

Grace [18] evaluated t% as a function of 8 for viscoelastic matrices. His results are shown in Fig. 10. The reduced time increases as the viscosity ratio increases. If the residence time during a processing is less than... 

A method for the estimation of composite material performance from the characteristics of fillers and the matrices and from the configuration of filler is generally called the law of mixture. In the most basic form of the law of mixture, the characteristics of a composite material are represented as a function of characteristics of constituent components and their volume fractions, as shown in Fig. 3. For a composite material (characteris-ticsiA f) that consists of component A (characteristics Xa, volume fraction ) and component B (characteristics Xf, volume fraction b), the basic formulae of the law of mixture are as follows ... 

There is greater similarity in the behavior of stretched melts and solid samples prepared by, e.g. pressure molding, probably, for the reason of parallelism in structure formation and destruction caused by deformation in melts and the amorphous regions of solid matrices. It is also possible to use identical equations for longitudinal viscosity and strength which present them as functions of the filler concentration [34]. 

An important tool in the study of matrices is provided by vector norms and matrix norms. A vector norm j... is any real valued function of the elements satisfying the following three conditions ... 

The methods of simple and of inverse iteration apply to arbitrary matrices, but many steps may be required to obtain sufficiently good convergence. It is, therefore, desirable to replace A, if possible, by a matrix that is similar (having the same roots) but having as many zeros as are reasonably obtainable in order that each step of the iteration require as few computations as possible. At the extreme, the characteristic polynomial itself could be obtained, but this is not necessarily advisable. The nature of the disadvantage can perhaps be made understandable from the following observation in the case of a full matrix, having no null elements, the n roots are functions of the n2 elements. They are also functions of the n coefficients of the characteristic equation, and cannot be expressed as functions of a smaller number of variables. It is to be expected, therefore, that they... 

Multiplication of Co-Representation Matrices.—We have referred above to the representations of nonunitary groups as co-representations. This distinction is made because the co-representation matrices for the group operators do not multiply in the same way as do the operators themselves.5 As will be seen below, this is a direct result of the fact that some of the operators in the group are antilinear. Consider that is the a 6 basis function of the i411 irreducible co-representation of G. The co-representation matrices D (u) and D (a) may be defined such that... 

Standardization. Standardization in analytical chemistry, in which standards are used to relate the instrument signal to compound concentration, is the critical function for determining the relative concentrations of species In a wide variety of matrices. Environmental Standard Reference Materials (SRM s) have been developed for various polynuclear aromatic hydrocarbons (PAH s). Information on SRM s can be obtained from the Office of Standard Reference Materials, National Bureau of Standards, Gaithersburg, MD 20899. Summarized in Table VII, these SRM s range from "pure compounds" in aqueous and organic solvents to "natural" matrices such as shale oil and urban and diesel particulate materials. 

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