Theorems · Theorem · linear algebra
Matrix.IsUnit.posSemidef_star_left_conjugate_iff
∀ {n : Type u_2} {R : Type u_3} [inst : Ring R] [inst_1 : PartialOrder R] [inst_2 : StarRing R] [inst_3 : Fintype n]
[inst_4 : DecidableEq n] {U x : Matrix n n R}, IsUnit U → ((star U * x * U).PosSemidef ↔ x.PosSemidef)For an invertible matrix U, star U * x * U is positive semi-definite iff x is.
This works on any ⋆-ring with a partial order.
See IsUnit.star_left_conjugate_nonneg_iff for a similar statement for star-ordered rings.
- Defined in
- Mathlib.LinearAlgebra.Matrix.PosDef
- Cited by
- 1 results in Mathlib
- Foundations
- Depth 82 from the axioms · uses propext, Classical.choice, Quot.sound
Around this declaration
Dashed lines are statement dependencies; solid lines are citations in proofs.
Cites19
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
- Ringstatement and proof · cited by 7,463
- PartialOrderstatement and proof · cited by 6,410
- Matrixstatement and proof · cited by 4,303
- mul_oneproof · cited by 3,885
- one_mulproof · cited by 2,841
- Unitsproof · cited by 2,804
- Units.valproof · cited by 1,966
- StarRingstatement and proof · cited by 1,686
- mul_assocproof · cited by 1,667
- IsUnitstatement and proof · cited by 1,602
- Star.starstatement and proof · cited by 1,082
Cited by1
Results whose statement or proof uses this declaration.
- Matrix.IsUnit.posSemidef_star_right_conjugate_iffproof · cited by 1