Square root of a matrix

The generation of the inverse square root of a matrix is a special case of the calculation of a general function of a matrix... 

Given a Hermitian matrix A, we can define a function of A, i.e., /(A), in much the same way we define functions f(x) of a simple variable x. For example, the square root of a matrix A, which we denote by A, is simply that matrix which when multiplied by itself gives A, i.e.,... 

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. 

Here RA is the response of the analyte at unit concentration, c E is a matrix of expected, or estimated, errors and F is the Froebus norm, or root sum of the squared elements, of a matrix. It should be noted that while the NAS is a matrix quantity, selectivity (SEL), sensitivity (SEN), and signal-to-noise (S/N) are all vector quantities. The limit of detection and the limit of quantitation can also be determined via any accepted univariate definition by substituting NAS P for the analyte signal and E P for the error value. 

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. 

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. 

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... 

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). 

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). 

---