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[Merged by Bors] - feat: Matrix.PosDef (A + B) and Matrix.PosSemidef (A + B) #13750

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[Merged by Bors] - feat: Matrix.PosDef (A + B) and Matrix.PosSemidef (A + B) #13750

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22 changes: 22 additions & 0 deletions Mathlib/LinearAlgebra/Matrix/PosDef.lean
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Expand Up @@ -123,6 +123,12 @@ protected lemma zpow [DecidableEq n] {M : Matrix n n R} (hM : M.PosSemidef) (z :
· simpa using hM.pow n
· simpa using (hM.pow n).inv

protected lemma add {A : Matrix m m R} {B : Matrix m m R}
(hA : A.PosSemidef) (hB : B.PosSemidef) : (A + B).PosSemidef :=
⟨hA.isHermitian.add hB.isHermitian, fun x => by
rw [add_mulVec, dotProduct_add]
exact add_nonneg (hA.2 x) (hB.2 x)⟩

/-- The eigenvalues of a positive semi-definite matrix are non-negative -/
lemma eigenvalues_nonneg [DecidableEq n] {A : Matrix n n 𝕜}
(hA : Matrix.PosSemidef A) (i : n) : 0 ≤ hA.1.eigenvalues i :=
Expand Down Expand Up @@ -315,6 +321,22 @@ theorem transpose {M : Matrix n n R} (hM : M.PosDef) : Mᵀ.PosDef := by
rw [mulVec_transpose, Matrix.dotProduct_mulVec, star_star, dotProduct_comm]
#align matrix.pos_def.transpose Matrix.PosDef.transpose

protected lemma add_posSemidef {A : Matrix m m R} {B : Matrix m m R}
(hA : A.PosDef) (hB : B.PosSemidef) : (A + B).PosDef :=
⟨hA.isHermitian.add hB.isHermitian, fun x hx => by
rw [add_mulVec, dotProduct_add]
exact add_pos_of_pos_of_nonneg (hA.2 x hx) (hB.2 x)⟩

protected lemma posSemidef_add {A : Matrix m m R} {B : Matrix m m R}
(hA : A.PosSemidef) (hB : B.PosDef) : (A + B).PosDef :=
⟨hA.isHermitian.add hB.isHermitian, fun x hx => by
rw [add_mulVec, dotProduct_add]
exact add_pos_of_nonneg_of_pos (hA.2 x) (hB.2 x hx)⟩

protected lemma add {A : Matrix m m R} {B : Matrix m m R}
(hA : A.PosDef) (hB : B.PosDef) : (A + B).PosDef :=
hA.add_posSemidef hB.posSemidef

theorem of_toQuadraticForm' [DecidableEq n] {M : Matrix n n ℝ} (hM : M.IsSymm)
(hMq : M.toQuadraticForm'.PosDef) : M.PosDef := by
refine ⟨hM, fun x hx => ?_⟩
Expand Down
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