Theorems · Theorem · linear algebra
mem_doublyStochastic_iff_sum
∀ {R : Type u_1} {n : Type u_2} [inst : Fintype n] [inst_1 : DecidableEq n] [inst_2 : Semiring R]
[inst_3 : PartialOrder R] [inst_4 : IsOrderedRing R] {M : Matrix n n R},
M ∈ doublyStochastic R n ↔ (∀ (i j : n), 0 ≤ M i j) ∧ (∀ (i : n), ∑ j, M i j = 1) ∧ ∀ (j : n), ∑ i, M i j = 1- Cited by
- 3 results in Mathlib
- Foundations
- Depth 84 from the axioms · uses propext, Classical.choice, Quot.sound
Around this declaration
Dashed lines are statement dependencies; solid lines are citations in proofs.
Cites17
Mathlib declarations this one mentions in its statement or cites explicitly in its proof. Plumbing is filtered out.
- Semiringstatement and proof · cited by 13,802
- Fintypestatement and proof · cited by 7,736
- PartialOrderstatement and proof · cited by 6,410
- Set.ofPredproof · cited by 6,101
- Finset.sumstatement and proof · cited by 5,195
- Matrixstatement and proof · cited by 4,303
- mul_oneproof · cited by 3,885
- Finset.univstatement and proof · cited by 3,473
- Submonoidstatement · cited by 3,086
- one_mulproof · cited by 2,841
- Finset.sum_congrproof · cited by 2,323
- IsOrderedRingstatement and proof · cited by 777
Cited by3
Results whose statement or proof uses this declaration.
- sum_row_of_mem_doublyStochasticproof · cited by 2
- sum_col_of_mem_doublyStochasticproof · cited by 0
- exists_mem_doublyStochastic_eq_smul_iffproof · cited by 0