Theorems · Theorem · linear algebra
Matrix.det_one_add_mul_comm
- 1000+ list: Sylvester's determinant theorem
∀ {m : Type u_2} {n : Type u_3} {α : Type u_4} [inst : Fintype m] [inst_1 : Fintype n] [inst_2 : DecidableEq m]
[inst_3 : DecidableEq n] [inst_4 : CommRing α] (A : Matrix m n α) (B : Matrix n m α),
(1 + A * B).det = (1 + B * A).detThe Weinstein–Aronszajn identity. Note the 1 on the LHS is of shape m×m, while the 1 on
the RHS is of shape n×n.
- Cited by
- 5 results in Mathlib
- Foundations
- Depth 95 from the axioms · uses propext, Classical.choice, Quot.sound
Around this declaration
Dashed lines are statement dependencies; solid lines are citations in proofs.
Cites9
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
- Matrix.detstatement and proof · cited by 665
- sub_neg_eq_addproof · cited by 264
- Matrix.neg_mulproof · cited by 8
- Matrix.mul_negproof · cited by 5
- Matrix.det_fromBlocks_one₁₁proof · cited by 1
- Matrix.det_fromBlocks_one₂₂proof · cited by 1
Cited by5
Results whose statement or proof uses this declaration.
- Matrix.det_one_sub_mul_commproof · cited by 1
- Matrix.det_mul_add_one_commproof · cited by 0
- Matrix.det_add_mulproof · cited by 0
- Matrix.det_add_replicateCol_mul_replicateRowproof · cited by 0
- Matrix.det_one_add_replicateCol_mul_replicateRowproof · cited by 0