JakobCHR.com

Quick Navigation:
 

Personal:

 Go to Home
 MS Research
 PhD Research
 Curriculum Vitae

General:

 Linux

Soon to come:

 Matlab
 On-line Stores
 Cycling
 Medicine & Health
 LaTeX
 OOP & C++
 Sony PCM-R500 DAT

Mathematical properties expressed using the matrix form

The matrix form of the equations (1.58) making up the discretized multi-group diffusion equations is a generalized eigenvalue problem since we may rewrite the matrix equation as

$$\nu \hspace{0.2ex}\underline{\underline{F}}{}\hspace{0.15ex... ...hspace{0.15ex} \hspace{0.2ex}\underline{\Phi}{}\hspace{0.15ex}$$ (1.43)

We are now able to identify that we are in fact dealing with a generalized eigenvalue problem since $\hspace{0.2ex}\underline{\underline{B}}{}\hspace{0.15ex} \ne \hspace{0.2ex}\underline{\underline{I}}{}\hspace{0.15ex}$.

We are, of course, able mathematically^1.20 to write the generalized eigenvalue problem in a form which resembles a simple eigenvalue problem. This is shown below

$$\nu \hspace{0.2ex}\underline{\underline{B}}{}\hspace{0.15ex... ...x} = \lambda_0 \hspace{0.2ex}\underline{\Phi}{}\hspace{0.15ex}$$ (1.44)

Finally, we are able to state the theorems specifying the mathematical properties of the discretized multi-group diffusion equations

Theorem 1.1 (Cf. Wachspress [4])   The real parts of the eigenvalues of the matrix $\hspace{0.2ex}\underline{\underline{B}}{}\hspace{0.15ex}$ are positive

$$Re[ \lambda(\hspace{0.2ex}\underline{\underline{B}}{}\hspace{0.15ex})] > 0$$ (1.45)

and the matrix $\hspace{0.2ex}\underline{\underline{B}}{}\hspace{0.15ex}$ has a non-negative inverse matrix $\hspace{0.2ex}\underline{\underline{B}}{}\hspace{0.15ex}^{-1}$.

Theorem 1.2 (Cf. Wachspress [4])   There is a unique positive eigenvector of the matrix $\nu \hspace{0.2ex}\underline{\underline{B}}{}\hspace{0.15ex}^{-1} \hspace{0.2ex}\underline{\underline{F}}{}\hspace{0.15ex}$, with real eigenvalue $\lambda_0$ greater in modulus than all other eigenvalues of matrix $\nu \hspace{0.2ex}\underline{\underline{B}}{}\hspace{0.15ex}^{-1} \hspace{0.2ex}\underline{\underline{F}}{}\hspace{0.15ex}$.

Theorem 1.3 (Cf. Wachspress [4])   For $\lambda_e > 0$, the matrix $(\hspace{0.2ex}\underline{\underline{B}}{}\hspace{0.15ex} - \frac{\nu}{\lambda_e}\hspace{0.2ex}\underline{\underline{F}}{}\hspace{0.15ex})$ has a non-negative inverse if and only if $\lambda_e > \lambda_0$.

Theorem 1.4 (Cf. Wachspress [4])   The inverse of any principal matrix minor^1.21 of $(\hspace{0.2ex}\underline{\underline{B}}{}\hspace{0.15ex} - \frac{\nu}{\lambda_e}\hspace{0.2ex}\underline{\underline{F}}{}\hspace{0.15ex})$ is non-negative for $\lambda_e > \lambda_0$^1.22.

Note that we in this mathematical treatment assume $\nu$ as being independent of energy--consequently we only have one $\nu$. This simplification is not by any means restrictive--the mathematical properties fit the situation of $\nu = \nu(E)$ as well since the energy dependence of $\nu$ only "perturbs" the $\overline{\Sigma}{}_{f,k}^g$'s slightly.

  Contents   Index  
 
 
 

Top Quality Developed with Danish Brain Power