misc2
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| misc2 [2013/12/06 16:39] – potthast | misc2 [2013/12/06 16:41] – potthast | ||
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| //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 69: | Line 70: | ||
| are influenced by different points, this means that there can be at most one nonzero entry | are influenced by different points, this means that there can be at most one nonzero entry | ||
| 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. | + | a permutation matrix $P$, with scaling $0$ allowed. |
| - | + | into a diagonal matrix, and the proof is complete | |
| - | $\Box$ \\ | + | |
misc2.txt · Last modified: 2023/03/28 09:14 by 127.0.0.1