Matrices square root

The information content resulting from both processing methods is identical insofar as correlation information is concerned. The matrix-square-root transformation can minimize artefacts due to relay effects and chemical shift near degeneracy (pseudo-relay effects80-82 98). The application of covariance methods to compute HSQC-1,1-ADEQUATE spectra is described in the following section. 

The diagonal elements of this matrix approximate the variances of the corresponding parameters. The square roots of these variances are estimates of the standard errors in the parameters and, in effect, are a measure of the uncertainties of those parameters. 

The simplest way to obtain X is to diagonalize S, take the reciprocal square roots of the eigenvalues and then transfomi the matrix back to its original representation, i.e. 

In sorjDtion experiments, the weight of sorbed molecules scales as tire square root of tire time, K4 t) ai t if diffusion obeys Pick s second law. Such behaviour is called case I diffusion. For some polymer/penetrant systems, M(t) is proportional to t. This situation is named case II diffusion [, ]. In tliese systems, sorjDtion strongly changes tire mechanical properties of tire polymers and a sharjD front of penetrant advances in tire polymer at a constant speed (figure C2.1.18). Intennediate behaviours between case I and case II have also been found. The occurrence of one mode, or tire otlier, is related to tire time tire polymer matrix needs to accommodate tire stmctural changes induced by tire progression of tire penetrant. 

L has only diagonal entries, the matrix L is identical to its transpose L = L. The atomic Drdinates are thus obtained by multiplying the square roots of the eigenvalues by the jenvectors. 

If two square matrices of the same size can be multiplied, then a square matrix can be multiplied into itself to obtain A, A, or A". A is the square root of A and the nth root of A". A number has only two square roots, but a matrix has infinitely many square roots. This will be demonstrated in the problems at the end of this chapter. 

VI. Finding Inverses, Square Roots, and Other Funetions of a Matrix Using its Eigenveetors and Eigenvalues... 

The two most common temporal input profiles for dmg delivery are zero order (constant release), and half order, ie, release that decreases with the square root of time. These two profiles correspond to diffusion through a membrane and desorption from a matrix, respectively (1,2). In practice, membrane systems have a period of constant release, ie, steady-state permeation, preceded by a period of either an increasing (time lag) or decreasing (burst) flux. This initial period may affect the time of appearance of a dmg in plasma on the first dose, but may become insignificant upon multiple dosing. 

The singular values of a complex n x m matrix A, denoted by cr,(A) are the nonnegative square-roots of the eigenvalues of A A ordered such that... 

Some coordinate transformations are non-linear, like transforming Cartesian to polar coordinates, where the polar coordinates are given in terms of square root and trigonometric functions of the Cartesian coordinates. This for example allows the Schrodinger equation for the hydrogen atom to be solved. Other transformations are linear, i.e. the new coordinate axes are linear combinations of the old coordinates. Such transfonnations can be used for reducing a matrix representation of an operator to a diagonal form. In the new coordinate system, the many-dimensional operator can be written as a sum of one-dimensional operators. 

It is first transfonned to mass-dependent coordinates by a G matrix eontaining the inverse square root of atomic masses (note that atomic, not nuclear, masses are used, this is in line with the Bom-Oppenheimer approximation that the electrons follow the nucleus). 

Matrix containing square root of inverse atomic masses... 

To compute the variance, we first find the mean concentration for that component over all of the samples. We then subtract this mean value from the concentration value of this component for each sample and square this difference. We then sum all of these squares and divide by the degrees of freedom (number of samples minus 1). The square root of the variance is the standard deviation. We adjust the variance to unity by dividing the concentration value of this component for each sample by the standard deviation. Finally, if we do not wish mean-centered data, we add back the mean concentrations that were initially subtracted. Equations [Cl] and [C2] show this procedure algebraically for component, k, held in a column-wise data matrix. 

The ordinary euclidean length is such a norm, and, more generally, if Q is any positive definite matrix, then the non-negative square root of... 

In vitro dissolution was virtually complete after 6-8 hr. Since the plot of cumulative drug release versus time is hyperbolic, the authors attempted to fit the data to the Higuchi matrix dissolution model (116,117), which predicts a linear correlation between cumulative drug release and the square root of time. Linearity occurred only between 20 and 70% release. 

The standard error of parameter ki,, is obtained as the square root of the corresponding diagonal element of the inverse of matrix A multiplied by 6g, i.e.,... 

The condition number is always greater than one and it represents the maximum amplification of the errors in the right hand side in the solution vector. The condition number is also equal to the square root of the ratio of the largest to the smallest singular value of A. In parameter estimation applications. A is a positive definite symmetric matrix and hence, the cond ) is also equal to the ratio of the largest to the smallest eigenvalue of A, i.e.,... 

The result shown in Eq. (54) is a square root of time relationship for moisture uptake. Mulski [20] demonstrated that for sodium glycinate in a hydrophobic porous matrix, moisture sorption follows Eq. (54). 

The initial rate at which the matrix C is formed in these matrix-dependent experiments is related to the initial concentration c by a square-root dependence. This square root law of autocatalysis is found in most self-replicating systems ... 

It is precisely this variance of (g) that we are after, because its square root gives us the angular dependent linewidth. A general expression in matrix notation can be derived for the variance (Hagen et al. 1985c) ... 

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