misc2
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| misc2 [2013/12/06 16:33] – potthast | misc2 [2023/03/28 09:14] (current) – external edit 127.0.0.1 | ||
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| located at $x_2$ and $f_{2}$ is only influenced by $\varphi_1$ located at $x_1$. The matrix | located at $x_2$ and $f_{2}$ is only influenced by $\varphi_1$ located at $x_1$. The matrix | ||
| \begin{equation} | \begin{equation} | ||
| + | \label{C example} | ||
| C = \left( \begin{array}{cc} c_{11} & c_{12} \\ c_{21} & c_{22} \end{array} \right) | C = \left( \begin{array}{cc} c_{11} & c_{12} \\ c_{21} & c_{22} \end{array} \right) | ||
| := \left( \begin{array}{cc} 1 & 0 \\ 3 & 0 \end{array} \right) | := \left( \begin{array}{cc} 1 & 0 \\ 3 & 0 \end{array} \right) | ||
| Line 52: | Line 53: | ||
| only. | only. | ||
| - | **Lemma.** | + | **Lemma.** |
| - | it can be transformed into a diagonal operator. | + | If we have a local operator |
| + | $x_{j} \in \mathbb{R}^d$, | ||
| + | it can be transformed into a diagonal operator. | ||
| + | output variables, diagonalization by reordering is not possible. | ||
| //Remark.// A reordering operation is equivalent to the application of a permutation | //Remark.// A reordering operation is equivalent to the application of a permutation | ||
| - | matrix $P$. | + | matrix $P$, i.e. a matrix which has exactly one element 1 in each row and column, with |
| + | all other elements zero. | ||
| //Proof.// We first assume that in the state space $X = \mathbb{R}^n$ each element belong | //Proof.// We first assume that in the state space $X = \mathbb{R}^n$ each element belong | ||
| Line 64: | Line 69: | ||
| is influenced by the entries in the $\ell$-th row of the matrix $H$. Thus, this is local | is influenced by the entries in the $\ell$-th row of the matrix $H$. Thus, this is local | ||
| if and only if at most one of the entries is non-zero. This applies to every row, i.e. in | if and only if at most one of the entries is non-zero. This applies to every row, i.e. in | ||
| - | each row there is at most one non-zero element. | + | each row there is at most one non-zero element. |
| - | + | are influenced by different points, this means that there can be at most one nonzero entry | |
| - | $\Box$ \\ | + | in each column as well. But that means that the operator $H$ looks like a scaled version of |
| + | a permutation matrix $P$, with scaling $0$ allowed. Clearly, by reordering we can make this | ||
| + | into a diagonal matrix. In general, we take (\ref{C example}) as counter example, | ||
| + | and the proof is complete | ||
misc2.1386344018.txt.gz · Last modified: 2023/03/28 09:14 (external edit)