Theorems · Theorem · linear algebra

sum_mulVec_of_mem_doublyStochastic

∀ {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} {x : n → R},
  M ∈ doublyStochastic R n → ∑ i, M.mulVec x i = ∑ i, x i

Applying a doubly stochastic matrix to a vector preserves its sum.

Defined in

Mathlib.Analysis.Convex.DoublyStochasticMatrix

Cited by

0 results in Mathlib

Foundations

Depth 87 from the axioms · uses propext, Classical.choice, Quot.sound

Assumes

FintypeDecidableEqSemiringPartialOrderIsOrderedRing