misc2

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misc2 [2013/12/06 16:41] potthast |
misc2 [2015/10/17 18:48] (current) |
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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) | ||

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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. In general, when a point influences two or more |

+ | 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 | ||

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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: 2015/10/17 18:48 (external edit)