Matrix inverse definition

Now we are ready to formulate the basic idea of the correction algorithm in order to correct the four-indexed operator f1, it is enough to correct the two-indexed operators fc and f 1 in the supermatrix representation (7.100). The real advantage of this proposal is its compatibility with any definite way of f2 and P7 correction [61, 294], The matrix inversion demanded in (7.99) is divided into two stages. In the fi, v subspace it is possible to find the inverse matrix analytically with the help of the Frobenius formula that is well known in matrix algebra [295]. The... 

SHELXL (Sheldrick and Schneider, 1997) is often viewed as a refinement program for high-resolution data only. Although it undoubtedly offers features needed for that resolution regime (optimization of anisotropic temperature factors, occupancy refinement, full matrix least squares to obtain standard deviations from the inverse Hessian matrix, flexible definitions for NCS, easiness to describe partially... 

More convenient forms of Eq. (54) that avoid matrix inversions are obtained after multiplication by the second-order moment matrix M [which is positive definite for linearly independent weight functions w,([Pg.291]

Newton s method and quasi-Newton techniques make use of second-order derivative information. Newton s method is computationally expensive because it requires analytical first-and second-order derivative information, as well as matrix inversion. Quasi-Newton methods rely on approximate second-order derivative information (Hessian) or an approximate Hessian inverse. There are a number of variants of these techniques from various researchers most quasi-Newton techniques attempt to find a Hessian matrix that is positive definite and well-conditioned at each iteration. Quasi-Newton methods are recognized as the most powerful unconstrained optimization methods currently available. 

For certain mathematical functions and operations it is necessary for the physicist to know their context, definition and mathematical properties, which we treat in the book. He does not need to know how to calculate them or to control their calculation. Numerical values of functions such as sinx have traditionally been taken from table books or slide rules. Modern computational facilities have enabled us to extend this concept, for example, to Coulomb functions, associated Legendre polynomials, Clebsch—Gordan and related coefficients, matrix inversion and diagonali-sation and Gaussian quadratures. The subroutine library has replaced the table book. We give references to suitable library subroutines. 

In this chapter we develop matrix algebra from two key perspectives one makes use of matrices to facilitate the handling of coordinate transformations, in preparation for a development of symmetry theory the other revisits determinants and, through the definition of the matrix inverse, provides a means for solving sets of linear equations. By the end of this chapter, you should ... 

In these methods, also known as quasi-Newton methods, the approximate Hessian is improved (updated) based on the results in previous steps. For the exact Hessian and a quadratic surface, the quasi-Newton equation = HAq and its analogue H Ag - = Aq - must hold (where Ag - = g - g and similarly for Aq - ). These equations, which have only n components, are obviously insufficient to determine the n(n + l)/2 independent components of the Hessian or its inverse. Therefore, the updating is arbitrary to a certain extent. It is desirable to have an updating scheme that converges to the exact Hessian for a quadratic function, preserves the quasi-Newton conditions obtained in previous steps, and—for minimization—keeps the Hessian positive definite. Updating can be performed on either F or its inverse, the approximate Hessian. In the former case repeated matrix inversion can be avoided. All updates use dyadic products, usually built... 

In the VSS, however, the back transformation from a chosen basis, like that represented by the collection attachable to the Jn( ) matrix, is not well defined, because the matrix inverses can no longer furnish the VSS basis set elements their columns belong to a VS in general, because the inverse matrix elements are no longer positive definite. For instance, the inverse of the Jn( ) matrix can be expressible as... 

Multibody hydrodynamic interactions have generally been ignored in simulations (for reasons of computational cost) with the notable exception of [243,264] for a monolayer system involving a small number of particles. Satoh et al. [266,267] approximate the multibody hydrodynamic forces by assuming additivity of the velocities. This does, however, not guarantee positive definiteness of the mobility matrix (inverse of the resistance matrix), imless a short cutoff radius of the hydrodynamic interactions is used [266,267]. 

The inverse dielectric matrix by definition satisfies the relations ... 

Adjugate Matrix of a Matrix Let Ay denote the cofactor of the element Oy in the determinant of the matrix A. The matrix B where B = (Ay) is called the adjugate matrix of A written adj A = B. The elements by are calculated by taking the matrix A, deleting the ith row and Jth. column, and calculating the determinant of the remaining matrix times (—1) Then A" = adj A/lAl. This definition may be used to calculate A"h However, it is very laborious and the inversion is usually accomplished by numerical techniques shown under Numerical Analysis and Approximate Methods. ... 

The effect of the specific values of the B j can be readily calculated for some simple laminates and can be calculated without significant difficulty for many more complex laminates. The influence of bending-extension coupling can be evaluated by use of the reduced bending stiffness approximation suggested by Ashton [7-20]. If you examine the matrix manipulations for the inversion of the force-strain-curvature and moment-strain-curvature relations (see Section 4.4), you will find a definition that relates to the reduced bending stiffness approximation. You will find that you could use as the bending stiffness of the entire structure,... 

The ri fiiatrix, due to the tune ordering operator in its definition is not invariant under time inversion. The invariance of the theory under tahi ihversidn has the following important consequence for the S-matrix since this operator s matrix elements axe given by ... 

But how can we tell then if the answer is correct Well, there is a way, and one that is not too overwhelming. From the definition of the inverse of a matrix, you should obtain a unit matrix if you multiply the inverse of a given matrix by the matrix itself. In our previous chapter [1] we showed this for the 2 x 2 case. For the simultaneous equations at hand, however, the process is only a little more extensive. From the original matrix of coefficients in the simultaneous equations that we are working with, the one called [A] above, we find that the inverse of this matrix is... 

In the same way, the parameters kij and hij are joined to form a unique parameter matrix H. With these definitions a linear problem may be written like that of equation 5. The matrix H can then be estimated either by direct pseudo-inversion or by PLS. It is worth noting... 

By definition, the transformation matrix M is full rank, and thus the inverse transformation... 

The covariance matrix is factored using the diagonal matrix A and the eigenvector matrix U as U UT. Since 5 is symmetric and positive-definite, the eigenvalues are positive and the eigenvectors orthogonal. The inverse S 1 of S can be expanded as UA 1UT and the transformation... 

Here m is the number of points in y and np the number of fitted parameters. The difference m-np is the number of degrees of freedom, df The elements djj in equation (4.32) are the diagonal elements of the inverse of the so-called curvature matrix, Curv, that contains the second derivatives of the sum of squares with respect to the parameters. The definition of the element Curvjk is... 

The preceding definition of a kinetic SDE reduces to that given by Hiitter and Ottinger [34] in the case of an invertible mobility matrix X P, for which Eq. (2.268) reduces to the requirement that Zap = K. In the case of a singular mobility, the present definition requires that the projection of Z p onto the nonnull subspace of K (corresponding to the soft subspace of a constrained system) equal the inverse of within this subspace, while leaving the components of Z p outside this subspace unspecified. 

EX14 1.4 Inversion of a positive definite symmetric matrix M16... 

Since A is symmetric, on input it is sufficient to store its corresponding portion in the lower triangular part of the array A, including the diagonal. The inverse is also symmetric, but an output it will occupy the entire matrix, since ithis i advantageous for further use. The zero-th row of array A is used as a vector of auxiliary variables, so do not store your own data here. If the matrix is not positive definite, the module will return the flag ER = 1. 

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