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Consequences of a reordering of unknowns

The four theorems mentioned above were all proven in a situation where the flux vector $\hspace{0.2ex}\underline{\Phi}{}\hspace{0.15ex}$ is ordered by group number g--this is seen most clearly in equation (1.49). The next question which arises is whether the theorems 1 through 4 also hold when the flux vector is ordered by the node number k. Let us for convenience write down the general eigenvalue problem once more

$$\lambda_0 \hspace{0.2ex}\underline{\underline{B}}{}\hspace{... ...hspace{0.15ex} \hspace{0.2ex}\underline{\Phi}{}\hspace{0.15ex}$$ (1.46)

where $\hspace{0.2ex}\underline{\underline{C}}{}\hspace{0.15ex}$ is defined by

$$\hspace{0.2ex}\underline{\underline{C}}{}\hspace{0.15ex} \;... ...\;\nu \hspace{0.2ex}\underline{\underline{F}}{}\hspace{0.15ex}$$ (1.47)

and $\Phi$ is ordered by group number g.

We now imagine ourselves to possess a permutation matrix^1.23, $\hspace{0.2ex}\underline{\underline{P}}{}\hspace{0.15ex}$, which interchanges rows of a matrix or column vector, ie changes the ordering in such a way that we end up with the wanted ordering. In this case we want $\hspace{0.2ex}\underline{\underline{P}}{}\hspace{0.15ex}$ to gather $\phi_k^g, \forall g$ that is $\phi_k^1$ and $\phi_k^2$ in a two group treatment.

We now multiply equation (1.64) with $\hspace{0.2ex}\underline{\underline{P}}{}\hspace{0.15ex}$ from the left and end up with

$$\lambda_0 \hspace{0.2ex}\underline{\underline{P}}{}\hspace{... ...hspace{0.15ex} \hspace{0.2ex}\underline{\Phi}{}\hspace{0.15ex}$$ (1.48)

Since it is always legitimate to multiply with the identity matrix $\hspace{0.2ex}\underline{\underline{I}}{}\hspace{0.15ex}$ we are now able to write the above mentioned equation as

$$\lambda_0 \hspace{0.2ex}\underline{\underline{P}}{}\hspace{... ...hspace{0.15ex} \hspace{0.2ex}\underline{\Phi}{}\hspace{0.15ex}$$ (1.49)

Now, by defining the following matrices and $\Phi$-vector

$$\begin{eqnarray*} \tilde{\hspace{0.2ex}\underline{\underline{B}}{}\hspace{0.15e... ...hspace{0.15ex} \hspace{0.2ex}\underline{\Phi}{}\hspace{0.15ex} \end{eqnarray*}$$

$\parbox{1.50cm}{\begin{eqnarray} \end{eqnarray}}$

we are able to rewrite (1.67) in the form

$$\lambda_0 \tilde{\hspace{0.2ex}\underline{\underline{B}}{}\... ...5ex}} \tilde{\hspace{0.2ex}\underline{\Phi}{}\hspace{0.15ex}}$$ (1.50)

This shows that the eigensolutions to the rewritten eigenvalue problem (1.69) are related to those of the original given by (1.64) in the following way

$\parbox{16cm}{ \begin{eqnarray*} \parb... ...{0.15ex} \hspace{0.2ex}\underline{\Phi}{}\hspace{0.15ex}$.} \end{eqnarray*}}$
Thus the proposed reordering only results in a reordering of the flux eigenvector leaving the eigenvalues unchanged, and we are able to conclude

Theorem 1.5   Any reordering of the unknown flux components still enables us to make use of Theorems 1 through 4 (with ( $\hspace{0.2ex}\underline{\underline{F}}{}\hspace{0.15ex},\hspace{0.2ex}\underline{\underline{B}}{}\hspace{0.15ex},\hspace{0.2ex}\underline{\Phi}{}\hspace{0.15ex}$) replaced by ( $\tilde{\hspace{0.2ex}\underline{\underline{F}}{}\hspace{0.15ex}},\tilde{\hspace... ...e{B}}{}\hspace{0.15ex}},\tilde{\hspace{0.2ex}\underline{\Phi}{}\hspace{0.15ex}}$)).

We may add that a permutation matrix has the following (nice) property

$$\hspace{0.2ex}\underline{\underline{P}}{}\hspace{0.15ex}^{-1} = \hspace{0.2ex}\underline{\underline{P}}{}\hspace{0.15ex}^T$$ (1.51)

This property makes it possible to state

$$\tilde{\hspace{0.2ex}\underline{\underline{B}}{}\hspace{0.1... ...x}\:\hspace{0.2ex}\underline{\underline{P}}{}\hspace{0.15ex}^T$$ (1.52)

which actually says that we get $\tilde{\hspace{0.2ex}\underline{\underline{B}}{}\hspace{0.15ex}}$ firstly by interchanging the rows in $\hspace{0.2ex}\underline{\underline{B}}{}\hspace{0.15ex}$ indicated by $\hspace{0.2ex}\underline{\underline{P}}{}\hspace{0.15ex}$ and subsequently interchanging the columns with numbers corresponding to the interchanged rows.

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