Theorems · Theorem · linear algebra
Matrix.posSemidef_conjTranspose_mul_self
∀ {m : Type u_1} {n : Type u_2} {R : Type u_3} [inst : Ring R] [inst_1 : PartialOrder R] [inst_2 : StarRing R]
[inst_3 : Fintype m] [Finite n] [StarOrderedRing R] (A : Matrix m n R), (A.conjTranspose * A).PosSemidefThe conjugate transpose of a matrix multiplied by the matrix is positive semidefinite
- Defined in
- Mathlib.LinearAlgebra.Matrix.PosDef
- Cited by
- 3 results in Mathlib
- Foundations
- Depth 79 from the axioms · uses propext, Classical.choice, Quot.sound
Around this declaration
Dashed lines are statement dependencies; solid lines are citations in proofs.
Cites22
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
- Finset.univproof · cited by 3,473
- Finitestatement and proof · cited by 3,029
- StarRingstatement and proof · cited by 1,686
- Star.starproof · cited by 1,082
- StarOrderedRingstatement and proof · cited by 587
- Matrix.mulVecproof · cited by 267
- Fintype.ofFiniteproof · cited by 255
- Matrix.conjTransposestatement and proof · cited by 202
Cited by3
Results whose statement or proof uses this declaration.
- Matrix.PosSemidef.kroneckerproof · cited by 2
- Matrix.posSemidef_self_mul_conjTransposeproof · cited by 1
- Matrix.eigenvalues_conjTranspose_mul_self_nonnegproof · cited by 0