Functions of a Matrix

Suppose we wish to compute /[[T]], /(A) being an arbitrary function of A. It can be shown that, for a large class of problems, there exists a function q(X) and a polynomial of degree m - 1, r(A), such that 

---

Find S 2 by using the program FUNCT MAT (this program generates a function of a matrix). This program will ask for the elements of the S array and write to disk a file... 

VI. Finding Inverses, Square Roots, and Other Functions of a Matrix Using its Eigenvectors and Eigenvalues... 

Now it can be shown that any function of a matrix A can be obtained by taking the same function of its corresponding diagonal alter ego and pre- and postmulti-plying by the diagonalizing matrix P and its inverse P-1 ... 

The norm function of a matrix is often useful and consists of the square root of the sum of squares, so in our example norm (W) equals 12.0419. This can be useful when scaling data, especially for vectors. Note that if Y is a row vector, then sqrt (Y Y ) is die same as norm(Y). 

The generation of the inverse square root of a matrix is a special case of the calculation of a general function of a matrix... 

If you will recall from the discussion of functions of a matrix in Chapter 1, we can form by diagonalizing S to form s, then taking the inverse square root of each of the eigenvalues to form the diagonal matrix s and then undiagonalizing by the transformation in (3.167). If S is Hermitian then is also Hermitian. Substituting (3.167) into (3.163),... 

Closing of mass balances is one way of checking the completeness of a model. However, the following independent test can also be used for linear models. It is based on the fact that the trace of an entire function of a matrix is equal to the sum of its eigenvalues,/(yli) + "/(iv,). In our case, the trace tr(t) of the matrix exp(kt) is then necessarily equal to the sum of all exponential decays corresponding to the eigenvalues of k with coefficients 1, that is. 

The evaluation of the matrix elements of the momentum-dependent operators is achieved using a theorem for functions of a matrix. If we have a matrix B whose eigenvalues b form a diagonal matrix represented by [fc] and whose eigenveetors are X, any function of the matrix can be expressed in terms of the function of the eigenvalues by... 

A major disadvantage of a matrix representation for a molecular graph is that the number of entries increases with the square of the number of atoms in the molecule. What is needed is a representation of a molecular graph where the number of entries increases only as a linear function of the number of atoms in the molecule. Such a representation can be obtained by listing, in tabular form only the atoms and the bonds of a molecular structure. In this case, the indices of the row and column of a matrix entry can be used for identifying an entry. In essence, one has to distinguish each atom and each bond in a molecule. This is achieved by a list of the atoms and a list of the bonds giving the coimections between the atoms. Such a representation is called a connection table (CT). 

Matrices obey an algebra of their own that resembles the algebra of ordinary numbers in some respects and not in others. The elements of a matrix may be numbers, operators, or functions. We shall deal primarily with matrices of numbers in this chapter, but matrices of operators and functions will be important later. 

This means that the dynamic functions of composites can be reduced to the dynamic functions of the matrix by a mere shift along the modulus axis by b(c). 

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... 

In a continuous game both the choice of strategy and the payoff as a function of that choice are continuous. The latter is particularly important because a discontinuous payoff function may not yield a solution. Thus, instead of a matrix [ow], a function M(x,y) gives the payoff each time a strategy is chosen (i.e., the value of x and y are fixed). The strategy of each player in this case is defined as a member of the class D of probability distribution functions that are defined as continuous, real-valued, monotonic functions such that... 

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 ... 

The most significant use for HTPB propints is in ballistic missiles (Refs 12 13). The most unique usage is in the functioning of a laser by means of the chemical exhaust species-generated by a HTPB-Xmm Perchlorate fuel matrix (Ref 9)... 

Here the relative intensities of the components of each branch are determined by the Boltzmann factor Correlation function K (t, J), corresponding to Gq(a>, J), is obviously the correlation function of a transition matrix element in Heisenberg representation... 

In the model advocated by Elliott et al. (2001) and George et al. (2003) the U-Th systematics are affected by partial melting leading to increases in ( Th/ Th). If the fluid does not contain appreciable Th (or Pa), then the slope of the U-Th array is simply a function of the matrix flow rate through the melting zone (Fig. 17a), rather than the time since U addition, and this bears similarities with the approach used to derive upwelling... 

One idea (not that we really do that) is to apply the Taylor series expansion on the exponential function of A, and evaluate the state transition matrix with... 

---