Exponential matrix function

We can find the solution to Eq. (4-1), which is simply a set of first order differential equations. As analogous to how Eq. (2-3) on page 2-2 was obtained, we now use the matrix exponential function as the integration factor, and the result is (hints in the Review Problems)... 

We next make use of Eq. (4-15) on page 4-14, i.e., the fact that we can expand the matrix exponential function as a closed-form series ... 

The matrix exponential function is herein defined in terms of an infinite series as in the scalar case... 

In this case, y and are vectors of the dependent variables and the initial conditions, respectively. The term e is the matrix exponential function, which can be obtained from Eq. (2.83) ... 

To conclude this section on two-compartment models we note that the hybrid constants a and p in the exponential function are eigenvalues of the matrix of coefficients of the system of linear differential equations ... 

In the general case there will be n roots which are the eigenvalues of the transfer matrix K. Each of the eigenvalues defines a particular phase of the time course of the contents in the n compartments of the model. The eigenvalues are the hybrid transfer constants which appear in the exponents of the exponential function. For example, for the ith compartment we obtain the general solution ... 

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... 

Instead of an infinite series, we can derive a closed form expression for the exponential function. For an n x n matrix A, we have... 

Such systems of differential equations are called homogeneous. They have as solutions, linear combinations of exponential functions, where the eigenvalues, Xi, of the matrix K are the exponentials. In the first, irreversible example, equation (5.34), the eigenvalues of K are Xi=-ki, X,2=-fe and X3=0. Thus, the concentration profiles are linear combinations of the vectors e-, where t is the vector of times. In matrix notation we can write... 

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... 

An approximate solution of Equation 22 is obtained from Equation 23 by suitably approximating the matrix exponential e K This is accomplished by the Fade approximants of the exponential function. These Fade approximants are rational functions of the form... 

The multiresponse counterpart of (1/covariance matrix 27 , which exists only if 27 has full rank. This condition is achievable by selecting a linearly independent set of responses, as described in Chapter 7. Then the exponential function in Eq. (4.2-11) may be generalized to... 

Using a similar experimental set-up as for the determination of Abraham s solvation parameters, the activity coefficient of solutes at infinite dilution y" cm be determined from their retention times using gas-liquid chromatography [12, 67-72], Alternatively, the diluter technique is applied [67, 73] for which an inert gas transports the solute from the headspace (which is in equilibrium with the ionic liquid matrix) to a GC-column. The continuous decrease of the concentration in the headspace is measured as a function of time, generating an exponential function from which y°° is calculated. 

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. 

The matrix exponential (exp(Ax)) is found as a function of the parameters and the independent variable (x) using the exponential(A,x) command in Maple. 

In this chapter semianalytical solutions (solutions analytical in t and numerical in x) were obtained for parabolic PDEs. In section 5.1.2, the given homogeneous parabolic PDE was converted to matrix form by applying finite differences in the spatial direction. The resulting matrix differential equation was then integrated analytically in time using Maple s matrix exponential. This methodology helps us solve the dependent variables at different node points as an analytical function of time. This is a powerful technique and is valid for all linear parabolic PDEs. This... 

The above computations were carried out with Eqs. 9.3.39 for the eigenvalues of the correction factor matrix. As noted earlier, this involves the computation of the error function that is significantly more time consuming than the exponential function needed for the film model correction factor. With the eigenvalues of [H] given by the film model... 

As shown in Equation 1, the capacity per cycle is directly proportional to the amount of antibody immobilized, the immobilization yield, the M.W. of the protein and the column volume and an exponential function of the number of cycles. The amount of antibody immobilized will usually be less than 10 gL l. Higher activation of the matrix required for greater than 10 gL l loading results in a decrease in the immobilization yield. The maximum immobilization yield is 1.0 (100%) while 0.8 (80%) is not difficult to obtain. The M.W. of the protein to be isolated is fixed. The only way to increase the capacity per cycle significantly is to increase the volume of the immunosorbent or increase the number of cycles prior to reaching 50% of initial capacity (cycle half-life). Increasing the volume of immunosorbent increases the amount of monoclonal antibody required. 

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