Exponential of a matrix

Equation (B2.4.13) is a pair of first-order differential equations, so its fonnal solution is given by equation (B2.4.14)), in which exp() means the exponential of a matrix. 

This has an exponential of a matrix. It is defined in terms of the expansion of the exponential function, see for example Smith [514, p. 134-5]. Now, the usual eigenvalue-eigenvector equation can be written in compact form,... 

As customary, the exponential of a matrix means the sum of the matrix series corresponding to the exponential function. The eigenvalues of i T) =e are called the Floquet multipliers. The eigenvalues of B are called the Floquet exponents. (There is some delicacy about the uniqueness of B which we will ignore because it is not relevant to our use.) Usually it is not possible to compute the Floquet exponents or multipliers. However, for low-dimensional systems of the kind we will investigate, there is a general theorem about the determinant of a fundamental matrix which is helpful. Let 4>(0 be a fundamental matrix for (4.1) with i (0) = I. Then... 

Moler and Van Loan (1978) discuss at some length Nineteen dubious ways to compute the exponential of a matrix. They consider three or four methods as candidates for the... 

In the final equality here we are using the weU-known Umit lim ., (1-1- xINf = e, which holds even when x is replaced by a matrix, here by tQ. The exponential of a matrix is defined by the Taylor... 

The exponential of a matrix can be defined, in the case of a matrix with a full set of eigenvectors, in terms of the eigenvector expansion. Alternatively, one may think of exp(A) as the sum of the exponential series... 

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

MvL78] Moler C. and van Loan C. (1978) Nineteen dubious way to compute the exponential of a matrix. SIAM Review 20(4) 801-836. 

For discrete-state variables, Eq. (5) simply requires exponentiation of a matrix, whereas for the case of continuous-state variables, it generally requires solution of the integral equation (4). 

The exponential of a diagonal matrix is again a diagonal matrix with exponentials of the diagonal elements, equation (B2.4.17)). 

Here we have an exponential of the matrix D. The matrix is zero except on the diagonal (containing the eigenvalues Aj, A2,..., A/y), and, as Smith proves, this and the definition of a matrix exponential lead to the simple result that... 

Using the Trotter approximation for the exponential of a sum of noncommuting operators allows separation of the kinetic and potential contributions to the propagation in each matrix element in the above expression, so that... 

The representation of this equation for anything greater than two variates is difficult to visualize, but the bivariate form (m = 2) serves to illustrate the general case. The exponential term in Equation (26) is of the form x Ax and is known as a quadratic form of a matrix product (Appendix A). Although the mathematical details associated with the quadratic form are not important for us here, one important property is that they have a well known geometric interpretation. All quadratic forms that occur in chemometrics and statistical data analysis expand to produce a quadratic smface that is a closed ellipse. Just as the univariate normal distribution appears bell-shaped, so the bivariate normal distribution is elliptical. 

The exponential matrix of a matrix can be obtained as follows > exponential(B,t) ... 

Find the matrix exponential of A (i.e., exp(At)) as a function of the parameters (rate constants) and the independent variable (t) using the Maple command exponential(A,t) . Call this matrix mat. 

Our method makes it possible to use simple linear rules for exploring complicated nonlinear systems. A simple application is the study of connectivity among various chemical species in complicated reaction networks. In the simple case of homogeneous systems with time-invariant structure, the susceptibility matrix x = [Xhh ] = X depends only on the transit time and not on time itself. The matrix elements Xuu ( ) are proportional to the elements (t) of a Green function matrix G (t) = [G / (t)], which is the exponential of a connectivity matrix K, that is, G (t) = exp [tK]. It follows that from a response experiment involving a system with time-invariant structure, it is possible to evaluate the connectivity matrix, K, which contains information about the relations among the different chemical species involved in the reaction mechanism. The nondiagonal elements of the matrix K = Kuu I show whether in the reaction mechanism there is a direct connection between two species in particular, if Kuu 0, there is a connection from the species u to the species u the reverse connection, from u io u, exists if Ku u 0. 

An alternative expression for the solution is in terms of the exponential of the matrix A scaled by time,... 

Solve the linear systems dz/dt = Az in the following special cases of 2 x 2 matrices A by direcdy computing the exponential of the matrix At ... 

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. 

P, where superscripts T and —1 represent the transpose and inverse of the matrix, respectively. This means, for real symmetric square matrices such as A here, exponential of the matrix At is given by... 

The solutions to equation (1) may be written down immediately in terms of a matrix exponential. For the two-dimensional NOESY experiment, with appropriate normalization, the initial conditions (at the beginning of the mixing period ) can be written as a unit matrix, i.e., the two-dimensional pulse sequence is equivalent to repeated relaxation experiments in which each spin in turn is displaced from equilibrium. The two-dimensional NOE cross-peak intensity at chemical shifts corresponding to spins i and j is then related to the magnetization of spin i for the experiment in which spin j was initially perturbed. After a mixing time tm, this is just exp(—Rrm),. ... 

Note that in MATLAB, the functions exp(A), c. o.r(A), sin A), are element-by-element functions and do not obey the above definitions. The MATLAB functions expm A), expmJ A), expm2 A), and expmSiA) calculate exponential of the matrix A by different algorithms. The function expm2(A) calculates exponential of the matrix A as in Eq. (2.83). 

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