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world:choquet [2026/04/10 15:48] rdouc [Krein-Milman, Choquet and Birkhoff-Von Neuman Theorems] |
world:choquet [2026/04/10 15:48] (current) rdouc [Krein-Milman, Choquet and Birkhoff-Von Neuman Theorems] |
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| **Theorem (Birkhoff–Von Neumann).** Let $B_n$ be the set of bistochastic matrices (of size $n\times n$), i.e. defined by | **Theorem (Birkhoff–Von Neumann).** Let $B_n$ be the set of bistochastic matrices (of size $n\times n$), i.e. defined by | ||
| $$ | $$ | ||
| - | B_n = \left\{ A = (a_{ij}) \in \mathcal{M}_n([0,1]) \mid \sum_j a_{ij} = 1,\ \sum_i a_{ij} = 1 \right\}. | + | B_n = \left\{ A = (a_{ij}) \in \mathcal{M}_n([0,1]) \mid \sum_{j=1}^n a_{ij} = 1,\ \sum_{i=1}^n a_{ij} = 1 \right\}. |
| $$ | $$ | ||
| Then $\mathrm{Extr}(B_n) = P_n$, the set of permutation matrices (i.e. matrices having exactly one $1$ in each row and each column). | Then $\mathrm{Extr}(B_n) = P_n$, the set of permutation matrices (i.e. matrices having exactly one $1$ in each row and each column). | ||
world/choquet.txt · Last modified: 2026/04/10 15:48 by rdouc
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