Theorems · Theorem · linear algebra
Matrix.PosSemidef.posDef_iff_isUnit
∀ {𝕜 : Type u_1} {n : Type u_2} [inst : RCLike 𝕜] [inst_1 : Fintype n] [inst_2 : DecidableEq n] {x : Matrix n n 𝕜},
x.PosSemidef → (x.PosDef ↔ IsUnit x)A positive semi-definite matrix is positive definite if and only if it is invertible.
- Defined in
- Mathlib.Analysis.Matrix.Order
- Cited by
- 3 results in Mathlib
- Foundations
- Depth 266 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.
Cites23
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
- RCLikestatement and proof · cited by 2,829
- IsUnitstatement and proof · cited by 1,602
- Star.starproof · cited by 1,082
- Matrix.mulVecproof · cited by 267
- Matrix.conjTransposeproof · cited by 202
- dotProductproof · cited by 194
- Matrix.vecMulproof · cited by 148
- RCLike.toPartialOrderstatement · cited by 106
- Matrix.PosSemidefstatement and proof · cited by 76
- Matrix.PosDefstatement and proof · cited by 68
Cited by3
Results whose statement or proof uses this declaration.
- Matrix.PosSemidef.posDef_iff_det_ne_zeroproof · cited by 2
- Matrix.isStrictlyPositive_iff_posDefproof · cited by 2
- Matrix.PosDef.kroneckerproof · cited by 1