misc2
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| Both sides previous revisionPrevious revisionNext revision | Previous revisionLast revisionBoth sides next revision | ||
| misc2 [2013/12/06 16:40] – potthast | misc2 [2013/12/06 16:45] – potthast | ||
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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 55: | Line 56: | ||
| If we have a local operator for which each measusment is influenced by a different point | If we have a local operator for which each measusment is influenced by a different point | ||
| $x_{j} \in \mathbb{R}^d$, | $x_{j} \in \mathbb{R}^d$, | ||
| - | it can be transformed into a diagonal operator. | + | 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 70: | Line 73: | ||
| in each column as well. But that means that the operator $H$ looks like a scaled version of | 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 | a permutation matrix $P$, with scaling $0$ allowed. Clearly, by reordering we can make this | ||
| - | into a diagonal matrix, and the proof is complete $\Box$ \\ | + | into a diagonal matrix. In general, we take (\ref{C example}) as counter example, |
| + | and the proof is complete $\Box$ \\ | ||
misc2.txt · Last modified: 2023/03/28 09:14 by 127.0.0.1