misc2
Differences
This shows you the differences between two versions of the page.
| Both sides previous revisionPrevious revisionNext revision | Previous revisionNext revisionBoth sides next revision | ||
| misc2 [2013/12/06 16:37] – potthast | misc2 [2013/12/06 16:41] – potthast | ||
|---|---|---|---|
| Line 58: | Line 58: | ||
| //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 66: | Line 67: | ||
| 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, and the proof is complete | ||
misc2.txt · Last modified: 2023/03/28 09:14 by 127.0.0.1