Matrix differentiation

Dijkwel, P.A. and Wenink, P.W. (1986). Structural integrity of the nuclear matrix differential effects of thiol agents and metal chelators. J. Cell Sci. 84, 53-67. 

Equation 6.9 is a matrix differential equation and represents a set of nxp ODEs. Once the sensitivity coefficients are obtained by solving numerically the above ODEs, the output vector, y(tl,k l+I ), can be computed. 

The parameter sensitivity matrix G(t) can be obtained as shown in the previous section by solving the matrix differential equation,... 

Similar to the parameter sensitivity matrix, the initial state sensitivity matrix, P(t), cannot be obtained by a simple differentiation. P(t) is determined by solving a matrix differential equation that is obtained by differentiating both sides of Equation 6.1 (state equation) with respect to p( ... 

LHSFs are determined at the center p of each shell. These LHSFs are then used to obtain the coupling matrix V i /nr(p p) given in Eq. (102). The coupled hyperradial equations in Eq. (101) are transformed into the coupled first-order nonlinear Bessel-Ricatti logarithmic matrix differential equation... 

Maronna, R. A. (1976). Robust M-estimators of multivariate location and scatter. Ann. Stat. 4, 51-67. Neudecker, H. (1969). Some theorems on matrix differentiation with special reference to Kronecker matrix products. Am. Stat. Assoc. J. 64,953—963. 

The evaluation of matrix elements for exphcitly correlated Gaussians (46) and (49) can be done in a very elegant and relatively simple way using matrix differential calculus. A systematic description of this very powerful mathematical tool is given in the book by Magnus and Neudecker [105]. The use of matrix differential calculus allows one to obtain compact expressions for matrix elements in the matrix form, which is very suitable for numerical computations [116,118] and perhaps facilitates a new theoretical insight. The present section is written in the spirit of Refs. 116 and 118, following most of the notation conventions therein. Thus, the reader can look for information about some basic ideas presented in these references if needed. 

It is known from matrix differential calculus that for a matrix variable X and a constant matrix C the following is true ... 

Here, we will need some simple facts from matrix differential calculus. If X is a matrix variable and (3 is a parameter that X depends on, then... 

We begin with the derivative of the secular equation with respect to energy eigenvalues. For some background on matrix differential calculus, see the Refs. 116 and 117. 

Matrix elements are scalar-valued matrix functions of the exponent matrices Lk- Therefore, the appropriate mathematical tool for finding derivatives is the matrix differential calculus [116, 118]. Using this, the derivations are nontrivial but straightforward. We will only present the final results of the derivations. The reader wishing to derive these formulas, or other matrix derivatives, is referred to the Ref. 116 and references therein. 

The gradients of the molecular integrals with respect to the nonlinear variational parameters (i.e., the exponential parameters Ak and the orbital centers Sk) were derived using the methods of matrix differential calculus as introduced by Kinghom [116]. It was shown there that the energy gradient with respect to all nonlinear variational parameters can be written as... 

J. R. Magnus and H. Neudecker, Matrix Differential Calculus with Applications in Statistics and Econometrics, Wiley, Chichester, 1988. 

Many chemical and biological processes are multistage. Multistage processes include absorption towers, distillation columns, and batteries of continuous stirred tank reactors (CSTRs). These processes may be either cocurrent or countercurrent. The steady state of a multistage process is usually described by a set of linear equations that can be treated via matrices. On the other hand, the unsteady-state dynamic behavior of a multistage process is usually described by a set of ordinary differential equations that gives rise to a matrix differential equation. 

The matrix/vector form of k, eqn.(34), allows us to exploit the powerful matrix differential calculus, described by Kinghorn[ll], for deriving elegant and easily implementable mathematical forms for integrals and their derivatives required in variational calculations. Alternatively, k can be written purely in terms of the vector variable r,... 

The power of the matrix differential calculus is immediately apparent when one actually computes an analytic gradient for a matrix function. The ease with which results are obtained and the concise compact form of the results seems almost miraculous at times. When the derivatives presented here where first formulated, the results were so surprising that numerical conformation was performed immediately. All of the following matrix derivatives have been confirmed by finite differences term by term on random matrices. 

D.B. Kinghorn, Explicitly Correlated Gaussian Basis Functions Derivation and Implementation of Matrix Elements and Gradient Formulas Using Matrix Differential Calculus, Ph.D. dissertation, Washington State University, 1995. 

We now present the solution of Eqs. (204) and (205) in terms of matrix continued fractions. The advantage of posing the problem in this way is that exact formulae in terms of such continued fractions may be written for the Laplace transform of the aftereffect function, the relaxation time, and the complex susceptibility. The starting point of the calculation is Eqs. (204) and (205) written as the matrix differential recurrence relation... 

Using the boundary conditions (equations (5.7) and (5.8)) the boundary values uo and Un+1 can be eliminated. Hence, the method of lines technique reduces the linear parabolic ODE partial differential equation (equation (5.1)) to a linear system of N coupled first order ordinary differential equations (equation (5.5)). Traditionally this linear system of ordinary differential equations is integrated numerically in time.[l] [2] [3] [4] However, since the governing equation (equation (5.5)) is linear, it can be written as a matrix differential equation (see section 2.1.2) ... 

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... 

When the principal components method was derived, matrix differentiation was used to determine the principal component vector p which minimized the sum of squared deviation. For this the symmetric mean centred variance-covariance matrix (X X) (X - X) was involved. 

Matrix differentiation was used in Appendix 3A to show that (X X) X> = b gives the least squares estimation of b. 

By assuming [D] to be constant the matrix differential equation may be uncoupled as described above and solved subject to the initial conditions ... 

In so far as [B] (or, more generally, [D] ) depends on the composition of the mixture and as the composition is, in turn, a function of position 17, we may regard [T ] as a function of 77. Thus, Eq. 8.3.50 is a first-order matrix differential equation of order n — 1 with a variable coefficient matrix [ (17)]. This equation may be solved by the method of repeated substitution as shown in Appendix B.2. The solution is... 

Krishna (1977) presented an approximate solution of Eqs. 8.7.3 by assuming that the coefficients and could be considered constant along the diffusion path. With these assumptions Eq. 8.7.3 represents a linear matrix differential equation, the solution of which can be written down in a manner exactly analogous to the ideal gas case. Thus, the composition profiles are given by... 

The solution to the matrix differential Eq. 10.4.6 can be found using the method of successive substitution (Appendix B.2). Here we follow closely the treatment by Taylor (1981b) (see, also Krishna, 1982). The solution to Eq. 10.4.6 can be written as... 

Equation 13.3.14 is a first-order matrix differential equation with constant coefficients (we have already assumed that [E] and [A/] are constant matrices). The solution is... 

In many cases, the solution to a matrix differential equation can be obtained as the matrix generalization of the equivalent scalar differential equation. For example, the first-order differential equation... 

The method of successive substitution is useful for solving matrix differential equations in which the coefficients are functions of the independent variable t. 

The same approach can be used to solve the matrix differential equation (Eq. B.2.1). The solution is (Amundson, 1966)... 

Let us demonstrate this method by reconsidering the matrix differential equation tZ(x)... 

An alternative to the methods described above can be used if the coefficient matrix is diagonalizable. Consider, once again, the matrix differential equation and its associated initial condition... 

The results of (1) and (2) apply to both Neel and Debye relaxation (see Section I.E). In each of the three cases it is shown how the longest relaxation time of the system may be obtained numerically by writing the set of differential-difference equations (called in the literature the Brink-man equations) as a matrix differential equation and successively increasing the size of the matrix until convergence is attained. It is also demonstrated how expressions for the relaxation times may be obtained by perturbation theory in the limit of low potential barriers. In Section... 

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