Theorems · Theorem · linear algebra
Matrix.IsHermitian.posSemidef_iff_eigenvalues_nonneg
∀ {n : Type u_2} {𝕜 : Type u_3} [inst : Fintype n] [inst_1 : RCLike 𝕜] {A : Matrix n n 𝕜} [inst_2 : DecidableEq n]
(hA : A.IsHermitian), A.PosSemidef ↔ 0 ≤ hA.eigenvaluesA Hermitian matrix is positive semi-definite if and only if its eigenvalues are non-negative.
- Defined in
- Mathlib.Analysis.Matrix.PosDef
- Cited by
- 2 results in Mathlib
- Foundations
- Depth 262 from the axioms · uses propext, Classical.choice, Quot.sound
- Assumes
- FintypeRCLikeDecidableEq
Around this declaration
Dashed lines are statement dependencies; solid lines are citations in proofs.
Cites21
Mathlib declarations this one mentions in its statement or cites explicitly in its proof. Plumbing is filtered out.
- DFunLike.coeproof · cited by 62,936
- Realstatement · cited by 25,697
- Fintypestatement and proof · cited by 7,736
- Ringproof · cited by 7,463
- PartialOrderproof · cited by 6,410
- Matrixstatement and proof · cited by 4,303
- RCLikestatement and proof · cited by 2,829
- StarRingproof · cited by 1,686
- RCLike.ofRealproof · cited by 350
- Matrix.diagonalproof · cited by 314
- unitaryproof · cited by 207
- StarMulproof · cited by 195
Cited by2
Results whose statement or proof uses this declaration.
- Matrix.PosSemidef.eigenvalues_nonnegproof · cited by 6
- Matrix.posSemidef_iff_isHermitian_and_spectrum_nonnegproof · cited by 0