Theorems · Theorem · linear algebra
Matrix.rank_mul_eq_left_of_isUnit_det
∀ {m : Type um} {n : Type un} [inst : Fintype n] {R : Type u_1} [inst_1 : CommRing R] [inst_2 : DecidableEq n]
(A : Matrix n n R) (B : Matrix m n R), IsUnit A.det → (B * A).rank = B.rankRight multiplying by an invertible matrix does not change the rank
- Defined in
- Mathlib.LinearAlgebra.Matrix.Rank
- Cited by
- 2 results in Mathlib
- Foundations
- Depth 121 from the axioms · uses propext, Classical.choice, Quot.sound
- Assumes
- FintypeCommRingDecidableEq
Around this declaration
Dashed lines are statement dependencies; solid lines are citations in proofs.
Cites8
Mathlib declarations this one mentions in its statement or cites explicitly in its proof. Plumbing is filtered out.
- CommRingstatement and proof · cited by 17,173
- Fintypestatement and proof · cited by 7,736
- Matrixstatement and proof · cited by 4,303
- IsUnitstatement and proof · cited by 1,602
- Matrix.detstatement and proof · cited by 665
- Matrix.rankstatement · cited by 48
- IsUnit.mem_nonZeroDivisorsproof · cited by 8
- Matrix.rank_mul_eq_left_of_det_mem_nonZeroDivisorsproof · cited by 2
Cited by2
Results whose statement or proof uses this declaration.
- Matrix.IsHermitian.rank_eq_rank_diagonalproof · cited by 1
- Matrix.exists_rank_normal_formproof · cited by 0