Theorems · Theorem · linear algebra
Matrix.rank_submatrix_le
∀ {m : Type um} {m₀ : Type um₀} {n : Type un} {n₀ : Type un₀} {R : Type uR} [inst : Fintype n] [inst_1 : CommSemiring R]
[StrongRankCondition R] [inst_3 : Fintype n₀] (A : Matrix m n R) (r : m₀ → m) (c : n₀ → n),
(A.submatrix r c).rank ≤ A.rankTaking a subset of the rows and columns reduces the rank.
- Defined in
- Mathlib.LinearAlgebra.Matrix.Rank
- Cited by
- 1 results in Mathlib
- Foundations
- Depth 101 from the axioms · uses propext, Classical.choice, Quot.sound
Around this declaration
Dashed lines are statement dependencies; solid lines are citations in proofs.
Cites45
Mathlib declarations this one mentions in its statement or cites explicitly in its proof. Plumbing is filtered out.
- DFunLike.coeproof · cited by 62,936
- RingHom.idproof · cited by 18,349
- CommSemiringstatement and proof · cited by 10,911
- LinearMapproof · cited by 10,215
- Fintypestatement and proof · cited by 7,736
- Submoduleproof · cited by 7,192
- Set.rangeproof · cited by 4,705
- Matrixstatement and proof · cited by 4,303
- Equiv.symmproof · cited by 3,681
- Nontrivialproof · cited by 2,416
- Module.finrankproof · cited by 1,770
- LinearMap.compproof · cited by 1,642
Cited by1
Results whose statement or proof uses this declaration.
- Matrix.IsPivotedBy.rank_eqproof · cited by 1