Positive Definite Matrices #

This file defines positive (semi)definite matrices and connects the notion to positive definiteness of quadratic forms. Most results require 𝕜 = ℝ or ℂ.

Main definitions #

Matrix.PosDef : a matrix M : Matrix n n 𝕜 is positive definite if it is Hermitian and xᴴMx is greater than zero for all nonzero x.
Matrix.PosSemidef : a matrix M : Matrix n n 𝕜 is positive semidefinite if it is Hermitian and xᴴMx is nonnegative for all x.

Main results #

Matrix.PosSemidef.fromBlocks₁₁ and Matrix.PosSemidef.fromBlocks₂₂: If a matrix A is positive definite, then [A B; Bᴴ D] is positive semidefinite if and only if D - Bᴴ A⁻¹ B is positive semidefinite.

Positive semidefinite matrices #

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def Matrix.PosSemidef {n : Type u_2} {R : Type u_3} [Fintype n] [Ring R] [PartialOrder R] [StarRing R] (M : Matrix n n R) :

Prop

A matrix M : Matrix n n R is positive semidefinite if it is Hermitian and xᴴ * M * x is nonnegative for all x.

Equations

M.PosSemidef = (M.IsHermitian ∧ ∀ (x : n → R), 0 ≤ star x ⬝ᵥ M.mulVec x)

Instances For

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theorem Matrix.PosSemidef.diagonal {n : Type u_2} {R : Type u_3} [Fintype n] [Ring R] [PartialOrder R] [StarRing R] [StarOrderedRing R] [DecidableEq n] {d : n → R} (h : 0 ≤ d) :

(diagonal d).PosSemidef

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theorem Matrix.posSemidef_diagonal_iff {n : Type u_2} {R : Type u_3} [Fintype n] [Ring R] [PartialOrder R] [StarRing R] [StarOrderedRing R] [DecidableEq n] {d : n → R} :

(diagonal d).PosSemidef ↔ ∀ (i : n), 0 ≤ d i

A diagonal matrix is positive semidefinite iff its diagonal entries are nonnegative.

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theorem Matrix.PosSemidef.isHermitian {n : Type u_2} {R : Type u_3} [Fintype n] [Ring R] [PartialOrder R] [StarRing R] {M : Matrix n n R} (hM : M.PosSemidef) :

M.IsHermitian

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theorem Matrix.PosSemidef.re_dotProduct_nonneg {n : Type u_2} {𝕜 : Type u_5} [Fintype n] [RCLike 𝕜] {M : Matrix n n 𝕜} (hM : M.PosSemidef) (x : n → 𝕜) :

0 ≤ RCLike.re (star x ⬝ᵥ M.mulVec x)

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theorem Matrix.PosSemidef.conjTranspose_mul_mul_same {n : Type u_2} {R : Type u_3} [Fintype n] [Ring R] [PartialOrder R] [StarRing R] {A : Matrix n n R} (hA : A.PosSemidef) {m : Type u_6} [Fintype m] (B : Matrix n m R) :

(B.conjTranspose * A * B).PosSemidef

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theorem Matrix.PosSemidef.mul_mul_conjTranspose_same {n : Type u_2} {R : Type u_3} [Fintype n] [Ring R] [PartialOrder R] [StarRing R] {A : Matrix n n R} (hA : A.PosSemidef) {m : Type u_6} [Fintype m] (B : Matrix m n R) :

(B * A * B.conjTranspose).PosSemidef

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theorem Matrix.PosSemidef.submatrix {m : Type u_1} {n : Type u_2} {R : Type u_3} [Fintype m] [Fintype n] [Ring R] [PartialOrder R] [StarRing R] {M : Matrix n n R} (hM : M.PosSemidef) (e : m → n) :

(M.submatrix e e).PosSemidef

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theorem Matrix.PosSemidef.transpose {n : Type u_2} {R' : Type u_4} [Fintype n] [CommRing R'] [PartialOrder R'] [StarRing R'] {M : Matrix n n R'} (hM : M.PosSemidef) :

M.transpose.PosSemidef

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@[simp]

theorem Matrix.posSemidef_transpose_iff {n : Type u_2} {R' : Type u_4} [Fintype n] [CommRing R'] [PartialOrder R'] [StarRing R'] {M : Matrix n n R'} :

M.transpose.PosSemidef ↔ M.PosSemidef

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theorem Matrix.PosSemidef.conjTranspose {n : Type u_2} {R : Type u_3} [Fintype n] [Ring R] [PartialOrder R] [StarRing R] {M : Matrix n n R} (hM : M.PosSemidef) :

M.conjTranspose.PosSemidef

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@[simp]

theorem Matrix.posSemidef_conjTranspose_iff {n : Type u_2} {R : Type u_3} [Fintype n] [Ring R] [PartialOrder R] [StarRing R] {M : Matrix n n R} :

M.conjTranspose.PosSemidef ↔ M.PosSemidef

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theorem Matrix.PosSemidef.zero {n : Type u_2} {R : Type u_3} [Fintype n] [Ring R] [PartialOrder R] [StarRing R] :

PosSemidef 0

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theorem Matrix.PosSemidef.one {n : Type u_2} {R : Type u_3} [Fintype n] [Ring R] [PartialOrder R] [StarRing R] [StarOrderedRing R] [DecidableEq n] :

PosSemidef 1

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theorem Matrix.PosSemidef.natCast {n : Type u_2} {R : Type u_3} [Fintype n] [Ring R] [PartialOrder R] [StarRing R] [StarOrderedRing R] [DecidableEq n] (d : ℕ) :

(↑d).PosSemidef

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theorem Matrix.PosSemidef.ofNat {n : Type u_2} {R : Type u_3} [Fintype n] [Ring R] [PartialOrder R] [StarRing R] [StarOrderedRing R] [DecidableEq n] (d : ℕ) [d.AtLeastTwo] :

(OfNat.ofNat d).PosSemidef

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theorem Matrix.PosSemidef.intCast {n : Type u_2} {R : Type u_3} [Fintype n] [Ring R] [PartialOrder R] [StarRing R] [StarOrderedRing R] [DecidableEq n] (d : ℤ) (hd : 0 ≤ d) :

(↑d).PosSemidef

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@[simp]

theorem Matrix.posSemidef_intCast_iff {n : Type u_2} {R : Type u_3} [Fintype n] [Ring R] [PartialOrder R] [StarRing R] [StarOrderedRing R] [DecidableEq n] [Nonempty n] [Nontrivial R] (d : ℤ) :

(↑d).PosSemidef ↔ 0 ≤ d

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theorem Matrix.PosSemidef.pow {n : Type u_2} {R : Type u_3} [Fintype n] [Ring R] [PartialOrder R] [StarRing R] [StarOrderedRing R] [DecidableEq n] {M : Matrix n n R} (hM : M.PosSemidef) (k : ℕ) :

(M ^ k).PosSemidef

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theorem Matrix.PosSemidef.inv {n : Type u_2} {R' : Type u_4} [Fintype n] [CommRing R'] [PartialOrder R'] [StarRing R'] [DecidableEq n] {M : Matrix n n R'} (hM : M.PosSemidef) :

M⁻¹.PosSemidef

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theorem Matrix.PosSemidef.zpow {n : Type u_2} {R' : Type u_4} [Fintype n] [CommRing R'] [PartialOrder R'] [StarRing R'] [StarOrderedRing R'] [DecidableEq n] {M : Matrix n n R'} (hM : M.PosSemidef) (z : ℤ) :

(M ^ z).PosSemidef

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theorem Matrix.PosSemidef.add {m : Type u_1} {R : Type u_3} [Fintype m] [Ring R] [PartialOrder R] [StarRing R] [AddLeftMono R] {A B : Matrix m m R} (hA : A.PosSemidef) (hB : B.PosSemidef) :

(A + B).PosSemidef

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theorem Matrix.PosSemidef.smul {n : Type u_2} {R : Type u_3} [Fintype n] [Ring R] [PartialOrder R] [StarRing R] {α : Type u_6} [CommSemiring α] [PartialOrder α] [StarRing α] [StarOrderedRing α] [Algebra α R] [StarModule α R] [PosSMulMono α R] {x : Matrix n n R} (hx : x.PosSemidef) {a : α} (ha : 0 ≤ a) :

(a • x).PosSemidef

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theorem Matrix.PosSemidef.eigenvalues_nonneg {n : Type u_2} {𝕜 : Type u_5} [Fintype n] [RCLike 𝕜] [DecidableEq n] {A : Matrix n n 𝕜} (hA : A.PosSemidef) (i : n) :

0 ≤ ⋯.eigenvalues i

The eigenvalues of a positive semi-definite matrix are non-negative

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theorem Matrix.PosSemidef.trace_nonneg {n : Type u_2} {𝕜 : Type u_5} [Fintype n] [RCLike 𝕜] {A : Matrix n n 𝕜} (hA : A.PosSemidef) :

0 ≤ A.trace

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theorem Matrix.PosSemidef.det_nonneg {n : Type u_2} {𝕜 : Type u_5} [Fintype n] [RCLike 𝕜] [DecidableEq n] {M : Matrix n n 𝕜} (hM : M.PosSemidef) :

0 ≤ M.det

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@[simp]

theorem Matrix.posSemidef_submatrix_equiv {m : Type u_1} {n : Type u_2} {R : Type u_3} [Fintype m] [Fintype n] [Ring R] [PartialOrder R] [StarRing R] {M : Matrix n n R} (e : m ≃ n) :

(M.submatrix ⇑e ⇑e).PosSemidef ↔ M.PosSemidef

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theorem Matrix.posSemidef_conjTranspose_mul_self {m : Type u_1} {n : Type u_2} {R : Type u_3} [Fintype m] [Fintype n] [Ring R] [PartialOrder R] [StarRing R] [StarOrderedRing R] (A : Matrix m n R) :

(A.conjTranspose * A).PosSemidef

The conjugate transpose of a matrix multiplied by the matrix is positive semidefinite

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theorem Matrix.posSemidef_self_mul_conjTranspose {m : Type u_1} {n : Type u_2} {R : Type u_3} [Fintype m] [Fintype n] [Ring R] [PartialOrder R] [StarRing R] [StarOrderedRing R] (A : Matrix m n R) :

(A * A.conjTranspose).PosSemidef

A matrix multiplied by its conjugate transpose is positive semidefinite

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theorem Matrix.posSemidef_sum {n : Type u_2} {R : Type u_3} [Fintype n] [Ring R] [PartialOrder R] [StarRing R] {ι : Type u_6} [AddLeftMono R] {x : ι → Matrix n n R} (s : Finset ι) (h : ∀ i ∈ s, (x i).PosSemidef) :

(∑ i ∈ s, x i).PosSemidef

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theorem Matrix.trace_conjTranspose_mul_self_eq_zero_iff {m : Type u_1} {n : Type u_2} [Fintype m] [Fintype n] {R : Type u_6} [PartialOrder R] [NonUnitalRing R] [StarRing R] [StarOrderedRing R] [NoZeroDivisors R] {A : Matrix m n R} :

(A.conjTranspose * A).trace = 0 ↔ A = 0

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theorem Matrix.trace_mul_conjTranspose_self_eq_zero_iff {m : Type u_1} {n : Type u_2} [Fintype m] [Fintype n] {R : Type u_6} [PartialOrder R] [NonUnitalRing R] [StarRing R] [StarOrderedRing R] [NoZeroDivisors R] {A : Matrix m n R} :

(A * A.conjTranspose).trace = 0 ↔ A = 0

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theorem Matrix.eigenvalues_conjTranspose_mul_self_nonneg {m : Type u_1} {n : Type u_2} {𝕜 : Type u_5} [Fintype m] [Fintype n] [RCLike 𝕜] (A : Matrix m n 𝕜) [DecidableEq n] (i : n) :

0 ≤ ⋯.eigenvalues i

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theorem Matrix.eigenvalues_self_mul_conjTranspose_nonneg {m : Type u_1} {n : Type u_2} {𝕜 : Type u_5} [Fintype m] [Fintype n] [RCLike 𝕜] (A : Matrix m n 𝕜) [DecidableEq m] (i : m) :

0 ≤ ⋯.eigenvalues i

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theorem Matrix.IsHermitian.posSemidef_iff_eigenvalues_nonneg {n : Type u_2} {𝕜 : Type u_5} [Fintype n] [RCLike 𝕜] [DecidableEq n] {A : Matrix n n 𝕜} (hA : A.IsHermitian) :

A.PosSemidef ↔ 0 ≤ hA.eigenvalues

A Hermitian matrix is positive semi-definite if and only if its eigenvalues are non-negative.

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@[deprecated Matrix.IsHermitian.posSemidef_iff_eigenvalues_nonneg (since := "2025-08-17")]

theorem Matrix.IsHermitian.posSemidef_of_eigenvalues_nonneg {n : Type u_2} {𝕜 : Type u_5} [Fintype n] [RCLike 𝕜] [DecidableEq n] {A : Matrix n n 𝕜} (hA : A.IsHermitian) :

0 ≤ hA.eigenvalues → A.PosSemidef

Alias of the reverse direction of Matrix.IsHermitian.posSemidef_iff_eigenvalues_nonneg.

A Hermitian matrix is positive semi-definite if and only if its eigenvalues are non-negative.

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theorem Matrix.PosSemidef.trace_eq_zero_iff {n : Type u_2} {𝕜 : Type u_5} [Fintype n] [RCLike 𝕜] {A : Matrix n n 𝕜} (hA : A.PosSemidef) :

A.trace = 0 ↔ A = 0

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theorem Matrix.IsUnit.posSemidef_conjugate_iff' {n : Type u_2} {R : Type u_3} [Fintype n] [Ring R] [PartialOrder R] [StarRing R] [DecidableEq n] {U x : Matrix n n R} (hU : 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.conjugate_nonneg_iff' for a similar statement for star-ordered rings.

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theorem Matrix.IsUnit.posSemidef_conjugate_iff {n : Type u_2} {R : Type u_3} [Fintype n] [Ring R] [PartialOrder R] [StarRing R] [DecidableEq n] {U x : Matrix n n R} (hU : IsUnit U) :

(U * x * star U).PosSemidef ↔ x.PosSemidef

For an invertible matrix U, U * x * star U is positive semi-definite iff x is. This works on any ⋆-ring with a partial order.

See IsUnit.conjugate_nonneg_iff for a similar statement for star-ordered rings.

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theorem Matrix.posSemidef_vecMulVec_self_star {n : Type u_2} {R : Type u_3} [Fintype n] [Ring R] [PartialOrder R] [StarRing R] [StarOrderedRing R] (a : n → R) :

(vecMulVec a (star a)).PosSemidef

The matrix vecMulVec a (star a) is always positive semi-definite.

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theorem Matrix.posSemidef_vecMulVec_star_self {n : Type u_2} {R : Type u_3} [Fintype n] [Ring R] [PartialOrder R] [StarRing R] [StarOrderedRing R] (a : n → R) :

(vecMulVec (star a) a).PosSemidef

The matrix vecMulVec (star a) a is always postive semi-definite.

Positive definite matrices #

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def Matrix.PosDef {n : Type u_2} {R : Type u_3} [Fintype n] [Ring R] [PartialOrder R] [StarRing R] (M : Matrix n n R) :

Prop

A matrix M : Matrix n n R is positive definite if it is Hermitian and xᴴMx is greater than zero for all nonzero x.

Equations

M.PosDef = (M.IsHermitian ∧ ∀ (x : n → R), x ≠ 0 → 0 < star x ⬝ᵥ M.mulVec x)

Instances For

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theorem Matrix.PosDef.isHermitian {n : Type u_2} {R : Type u_3} [Fintype n] [Ring R] [PartialOrder R] [StarRing R] {M : Matrix n n R} (hM : M.PosDef) :

M.IsHermitian

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theorem Matrix.PosDef.re_dotProduct_pos {n : Type u_2} {𝕜 : Type u_5} [Fintype n] [RCLike 𝕜] {M : Matrix n n 𝕜} (hM : M.PosDef) {x : n → 𝕜} (hx : x ≠ 0) :

0 < RCLike.re (star x ⬝ᵥ M.mulVec x)

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theorem Matrix.PosDef.posSemidef {n : Type u_2} {R : Type u_3} [Fintype n] [Ring R] [PartialOrder R] [StarRing R] {M : Matrix n n R} (hM : M.PosDef) :

M.PosSemidef

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theorem Matrix.PosDef.transpose {n : Type u_2} {R' : Type u_4} [Fintype n] [CommRing R'] [PartialOrder R'] [StarRing R'] {M : Matrix n n R'} (hM : M.PosDef) :

M.transpose.PosDef

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@[simp]

theorem Matrix.PosDef.transpose_iff {n : Type u_2} {R' : Type u_4} [Fintype n] [CommRing R'] [PartialOrder R'] [StarRing R'] {M : Matrix n n R'} :

M.transpose.PosDef ↔ M.PosDef

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theorem Matrix.PosDef.diagonal {n : Type u_2} {R : Type u_3} [Fintype n] [Ring R] [PartialOrder R] [StarRing R] [StarOrderedRing R] [DecidableEq n] [NoZeroDivisors R] {d : n → R} (h : ∀ (i : n), 0 < d i) :

(diagonal d).PosDef

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@[simp]

theorem Matrix.posDef_diagonal_iff {n : Type u_2} {R : Type u_3} [Fintype n] [Ring R] [PartialOrder R] [StarRing R] [StarOrderedRing R] [DecidableEq n] [NoZeroDivisors R] [Nontrivial R] {d : n → R} :

(diagonal d).PosDef ↔ ∀ (i : n), 0 < d i

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theorem Matrix.PosDef.one {n : Type u_2} {R : Type u_3} [Fintype n] [Ring R] [PartialOrder R] [StarRing R] [StarOrderedRing R] [DecidableEq n] [NoZeroDivisors R] :

PosDef 1

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theorem Matrix.PosDef.natCast {n : Type u_2} {R : Type u_3} [Fintype n] [Ring R] [PartialOrder R] [StarRing R] [StarOrderedRing R] [DecidableEq n] [NoZeroDivisors R] (d : ℕ) (hd : d ≠ 0) :

(↑d).PosDef

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@[simp]

theorem Matrix.posDef_natCast_iff {n : Type u_2} {R : Type u_3} [Fintype n] [Ring R] [PartialOrder R] [StarRing R] [StarOrderedRing R] [DecidableEq n] [NoZeroDivisors R] [Nonempty n] [Nontrivial R] {d : ℕ} :

(↑d).PosDef ↔ 0 < d

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theorem Matrix.PosDef.ofNat {n : Type u_2} {R : Type u_3} [Fintype n] [Ring R] [PartialOrder R] [StarRing R] [StarOrderedRing R] [DecidableEq n] [NoZeroDivisors R] (d : ℕ) [d.AtLeastTwo] :

(OfNat.ofNat d).PosDef

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theorem Matrix.PosDef.intCast {n : Type u_2} {R : Type u_3} [Fintype n] [Ring R] [PartialOrder R] [StarRing R] [StarOrderedRing R] [DecidableEq n] [NoZeroDivisors R] (d : ℤ) (hd : 0 < d) :

(↑d).PosDef

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@[simp]

theorem Matrix.posDef_intCast_iff {n : Type u_2} {R : Type u_3} [Fintype n] [Ring R] [PartialOrder R] [StarRing R] [StarOrderedRing R] [DecidableEq n] [NoZeroDivisors R] [Nonempty n] [Nontrivial R] {d : ℤ} :

(↑d).PosDef ↔ 0 < d

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theorem Matrix.PosDef.add_posSemidef {m : Type u_1} {R : Type u_3} [Fintype m] [Ring R] [PartialOrder R] [StarRing R] [AddLeftMono R] {A B : Matrix m m R} (hA : A.PosDef) (hB : B.PosSemidef) :

(A + B).PosDef

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theorem Matrix.PosDef.posSemidef_add {m : Type u_1} {R : Type u_3} [Fintype m] [Ring R] [PartialOrder R] [StarRing R] [AddLeftMono R] {A B : Matrix m m R} (hA : A.PosSemidef) (hB : B.PosDef) :

(A + B).PosDef

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theorem Matrix.PosDef.add {m : Type u_1} {R : Type u_3} [Fintype m] [Ring R] [PartialOrder R] [StarRing R] [AddLeftMono R] {A B : Matrix m m R} (hA : A.PosDef) (hB : B.PosDef) :

(A + B).PosDef

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theorem Matrix.posDef_sum {m : Type u_1} {R : Type u_3} [Fintype m] [Ring R] [PartialOrder R] [StarRing R] {ι : Type u_6} [AddLeftMono R] {A : ι → Matrix m m R} {s : Finset ι} (hs : s.Nonempty) (hA : ∀ i ∈ s, (A i).PosDef) :

(∑ i ∈ s, A i).PosDef

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theorem Matrix.PosDef.smul {n : Type u_2} {R : Type u_3} [Fintype n] [Ring R] [PartialOrder R] [StarRing R] {α : Type u_6} [CommSemiring α] [PartialOrder α] [StarRing α] [StarOrderedRing α] [Algebra α R] [StarModule α R] [PosSMulStrictMono α R] {x : Matrix n n R} (hx : x.PosDef) {a : α} (ha : 0 < a) :

(a • x).PosDef

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theorem Matrix.PosDef.conjTranspose_mul_mul_same {m : Type u_1} {n : Type u_2} {R : Type u_3} [Fintype m] [Fintype n] [Ring R] [PartialOrder R] [StarRing R] {A : Matrix n n R} {B : Matrix n m R} (hA : A.PosDef) (hB : Function.Injective B.mulVec) :

(B.conjTranspose * A * B).PosDef

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theorem Matrix.PosDef.mul_mul_conjTranspose_same {m : Type u_1} {n : Type u_2} {R : Type u_3} [Fintype m] [Fintype n] [Ring R] [PartialOrder R] [StarRing R] {A : Matrix n n R} {B : Matrix m n R} (hA : A.PosDef) (hB : Function.Injective fun (v : m → R) => vecMul v B) :

(B * A * B.conjTranspose).PosDef

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theorem Matrix.PosDef.conjTranspose_mul_self {m : Type u_1} {n : Type u_2} {R : Type u_3} [Fintype m] [Fintype n] [Ring R] [PartialOrder R] [StarRing R] [StarOrderedRing R] [NoZeroDivisors R] (A : Matrix m n R) (hA : Function.Injective A.mulVec) :

(A.conjTranspose * A).PosDef

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theorem Matrix.PosDef.mul_conjTranspose_self {m : Type u_1} {n : Type u_2} {R : Type u_3} [Fintype m] [Fintype n] [Ring R] [PartialOrder R] [StarRing R] [StarOrderedRing R] [NoZeroDivisors R] (A : Matrix m n R) (hA : Function.Injective fun (v : m → R) => vecMul v A) :

(A * A.conjTranspose).PosDef

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theorem Matrix.PosDef.conjTranspose {n : Type u_2} {R : Type u_3} [Fintype n] [Ring R] [PartialOrder R] [StarRing R] {M : Matrix n n R} (hM : M.PosDef) :

M.conjTranspose.PosDef

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@[simp]

theorem Matrix.posDef_conjTranspose_iff {n : Type u_2} {R : Type u_3} [Fintype n] [Ring R] [PartialOrder R] [StarRing R] {M : Matrix n n R} :

M.conjTranspose.PosDef ↔ M.PosDef

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theorem Matrix.PosDef.of_toQuadraticForm' {n : Type u_2} [Fintype n] [DecidableEq n] {M : Matrix n n ℝ} (hM : M.IsSymm) (hMq : M.toQuadraticMap'.PosDef) :

M.PosDef

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theorem Matrix.PosDef.toQuadraticForm' {n : Type u_2} [Fintype n] [DecidableEq n] {M : Matrix n n ℝ} (hM : M.PosDef) :

M.toQuadraticMap'.PosDef

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theorem Matrix.PosDef.eigenvalues_pos {n : Type u_2} {𝕜 : Type u_5} [Fintype n] [RCLike 𝕜] [DecidableEq n] {A : Matrix n n 𝕜} (hA : A.PosDef) (i : n) :

0 < ⋯.eigenvalues i

The eigenvalues of a positive definite matrix are positive

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theorem Matrix.IsHermitian.posDef_iff_eigenvalues_pos {n : Type u_2} {𝕜 : Type u_5} [Fintype n] [RCLike 𝕜] [DecidableEq n] {A : Matrix n n 𝕜} (hA : A.IsHermitian) :

A.PosDef ↔ ∀ (i : n), 0 < hA.eigenvalues i

A Hermitian matrix is positive-definite if and only if its eigenvalues are positive.

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theorem Matrix.PosDef.trace_pos {n : Type u_2} {𝕜 : Type u_5} [Fintype n] [RCLike 𝕜] [Nonempty n] {A : Matrix n n 𝕜} (hA : A.PosDef) :

0 < A.trace

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theorem Matrix.PosDef.det_pos {n : Type u_2} {𝕜 : Type u_5} [Fintype n] [RCLike 𝕜] [DecidableEq n] {M : Matrix n n 𝕜} (hM : M.PosDef) :

0 < M.det

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theorem Matrix.PosDef.isUnit {n : Type u_2} [Fintype n] {K : Type u_6} [Field K] [PartialOrder K] [StarRing K] [DecidableEq n] {M : Matrix n n K} (hM : M.PosDef) :

IsUnit M

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theorem Matrix.PosDef.inv {n : Type u_2} [Fintype n] {K : Type u_6} [Field K] [PartialOrder K] [StarRing K] [DecidableEq n] {M : Matrix n n K} (hM : M.PosDef) :

M⁻¹.PosDef

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@[simp]

theorem Matrix.posDef_inv_iff {n : Type u_2} [Fintype n] {K : Type u_6} [Field K] [PartialOrder K] [StarRing K] [DecidableEq n] {M : Matrix n n K} :

M⁻¹.PosDef ↔ M.PosDef

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theorem Matrix.IsUnit.posDef_conjugate_iff' {n : Type u_2} {R : Type u_3} [Fintype n] [Ring R] [PartialOrder R] [StarRing R] [DecidableEq n] {x U : Matrix n n R} (hU : IsUnit U) :

(star U * x * U).PosDef ↔ x.PosDef

For an invertible matrix U, star U * x * U is positive definite iff x is. This works on any ⋆-ring with a partial order.

See IsUnit.isStrictlyPositive_conjugate_iff' for a similar statement for star-ordered rings. For matrices, positive definiteness is equivalent to strict positivity when the underlying field is ℝ or ℂ (see Matrix.isStrictlyPositive_iff_posDef).

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theorem Matrix.IsUnit.posDef_conjugate_iff {n : Type u_2} {R : Type u_3} [Fintype n] [Ring R] [PartialOrder R] [StarRing R] [DecidableEq n] {x U : Matrix n n R} (hU : IsUnit U) :

(U * x * star U).PosDef ↔ x.PosDef

For an invertible matrix U, U * x * star U is positive definite iff x is. This works on any ⋆-ring with a partial order.

See IsUnit.isStrictlyPositive_conjugate_iff for a similar statement for star-ordered rings. For matrices, positive definiteness is equivalent to strict positivity when the underlying field is ℝ or ℂ (see Matrix.isStrictlyPositive_iff_posDef).

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