Theorems · Theorem · linear algebra
Matrix.PosSemidef.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.PosSemidef) (i : n), 0 ≤ ⋯.eigenvalues iThe eigenvalues of a positive semi-definite matrix are non-negative
- Defined in
- Mathlib.Analysis.Matrix.PosDef
- Cited by
- 6 results in Mathlib
- Foundations
- Depth 263 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.
Cites13
Mathlib declarations this one mentions in its statement or cites explicitly in its proof. Plumbing is filtered out.
- Realstatement · cited by 25,697
- Fintypestatement and proof · cited by 7,736
- Finsuppstatement · cited by 5,255
- Matrixstatement and proof · cited by 4,303
- RCLikestatement and proof · cited by 2,829
- Star.starstatement · cited by 1,082
- Finsupp.sumstatement · cited by 481
- Matrix.IsHermitianstatement · cited by 126
- RCLike.toPartialOrderstatement · cited by 106
- Matrix.PosSemidefstatement and proof · cited by 76
- Matrix.IsHermitian.eigenvaluesstatement · cited by 34
- Matrix.PosSemidef.isHermitianproof · cited by 8
Cited by6
Results whose statement or proof uses this declaration.
- Matrix.PosSemidef.det_nonnegproof · cited by 1
- Matrix.eigenvalues_conjTranspose_mul_self_nonnegproof · cited by 0
- Matrix.eigenvalues_self_mul_conjTranspose_nonnegproof · cited by 0
- Matrix.posSemidef_iff_isHermitian_and_spectrum_nonnegproof · cited by 0
- Matrix.PosSemidef.trace_eq_zero_iffproof · cited by 0
- Matrix.PosSemidef.det_sqrtproof · cited by 0