Theorems · Theorem · linear algebra
Matrix.IsHermitian.det_eq_prod_eigenvalues
∀ {𝕜 : Type u_1} [inst : RCLike 𝕜] {n : Type u_2} [inst_1 : Fintype n] {A : Matrix n n 𝕜} [inst_2 : DecidableEq n]
(hA : A.IsHermitian), A.det = ∏ i, ↑(hA.eigenvalues i)The determinant of a Hermitian matrix is the product of its eigenvalues.
- Defined in
- Mathlib.Analysis.Matrix.Spectrum
- Cited by
- 4 results in Mathlib
- Foundations
- Depth 265 from the axioms · uses propext, Classical.choice, Quot.sound
- Assumes
- RCLikeFintypeDecidableEq
Around this declaration
Dashed lines are statement dependencies; solid lines are citations in proofs.
Cites15
Mathlib declarations this one mentions in its statement or cites explicitly in its proof. Plumbing is filtered out.
- Fintypestatement and proof · cited by 7,736
- Matrixstatement and proof · cited by 4,303
- Finset.univstatement and proof · cited by 3,473
- RCLikestatement and proof · cited by 2,829
- Finset.prodstatement and proof · cited by 2,356
- Matrix.detstatement · cited by 665
- Multiset.prodproof · cited by 528
- Finset.valproof · cited by 438
- RCLike.ofRealstatement and proof · cited by 350
- Multiset.map_congrproof · cited by 232
- Matrix.IsHermitianstatement and proof · cited by 126
- Matrix.IsHermitian.eigenvaluesstatement and proof · cited by 34
Cited by4
Results whose statement or proof uses this declaration.
- Matrix.PosSemidef.det_nonnegproof · cited by 1
- Matrix.PosSemidef.det_sqrtproof · cited by 0
- Matrix.IsHermitian.det_absproof · cited by 0
- Matrix.PosDef.det_posproof · cited by 0