Documentation

Mathlib.LinearAlgebra.Matrix.SchurComplement

2×2 block matrices and the Schur complement #

This file proves properties of 2×2 block matrices [A B; C D] that relate to the Schur complement D - C*A⁻¹*B.

Some of the results here generalize to 2×2 matrices in a category, rather than just a ring. A few results in this direction can be found in Mathlib/CategoryTheory/Preadditive/Biproducts.lean, especially the declarations CategoryTheory.Biprod.gaussian and CategoryTheory.Biprod.isoElim. Compare with Matrix.invertibleOfFromBlocks₁₁Invertible.

Main results #

Matrix.det_fromBlocks₁₁, Matrix.det_fromBlocks₂₂: determinant of a block matrix in terms of the Schur complement.
Matrix.invOf_fromBlocks_zero₂₁_eq, Matrix.invOf_fromBlocks_zero₁₂_eq: the inverse of a block triangular matrix.
Matrix.isUnit_fromBlocks_zero₂₁, Matrix.isUnit_fromBlocks_zero₁₂: invertibility of a block triangular matrix.
Matrix.det_one_add_mul_comm: the Weinstein–Aronszajn identity.
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.

theorem Matrix.fromBlocks_eq_of_invertible₁₁ {l : Type u_1} {m : Type u_2} {n : Type u_3} {α : Type u_4} [Fintype l] [Fintype m] [Fintype n] [DecidableEq l] [DecidableEq m] [DecidableEq n] [CommRing α] (A : Matrix m m α) (B : Matrix m n α) (C : Matrix l m α) (D : Matrix l n α) [Invertible A] :

fromBlocks A B C D = fromBlocks 1 0 (C * ⅟ A) 1 * fromBlocks A 0 0 (D - C * ⅟ A * B) * fromBlocks 1 (⅟ A * B) 0 1

LDU decomposition of a block matrix with an invertible top-left corner, using the Schur complement.

theorem Matrix.fromBlocks_eq_of_invertible₂₂ {l : Type u_1} {m : Type u_2} {n : Type u_3} {α : Type u_4} [Fintype l] [Fintype m] [Fintype n] [DecidableEq l] [DecidableEq m] [DecidableEq n] [CommRing α] (A : Matrix l m α) (B : Matrix l n α) (C : Matrix n m α) (D : Matrix n n α) [Invertible D] :

fromBlocks A B C D = fromBlocks 1 (B * ⅟ D) 0 1 * fromBlocks (A - B * ⅟ D * C) 0 0 D * fromBlocks 1 0 (⅟ D * C) 1

LDU decomposition of a block matrix with an invertible bottom-right corner, using the Schur complement.

Block triangular matrices #

def Matrix.fromBlocksZero₂₁Invertible {m : Type u_2} {n : Type u_3} {α : Type u_4} [Fintype m] [Fintype n] [DecidableEq m] [DecidableEq n] [CommRing α] (A : Matrix m m α) (B : Matrix m n α) (D : Matrix n n α) [Invertible A] [Invertible D] :

Invertible (fromBlocks A B 0 D)

An upper-block-triangular matrix is invertible if its diagonal is.

Equations

A.fromBlocksZero₂₁Invertible B D = (Matrix.fromBlocks A B 0 D).invertibleOfLeftInverse (Matrix.fromBlocks (⅟ A) (-(⅟ A * B * ⅟ D)) 0 ⅟ D) ⋯

def Matrix.fromBlocksZero₁₂Invertible {m : Type u_2} {n : Type u_3} {α : Type u_4} [Fintype m] [Fintype n] [DecidableEq m] [DecidableEq n] [CommRing α] (A : Matrix m m α) (C : Matrix n m α) (D : Matrix n n α) [Invertible A] [Invertible D] :

Invertible (fromBlocks A 0 C D)

A lower-block-triangular matrix is invertible if its diagonal is.

Equations

A.fromBlocksZero₁₂Invertible C D = (Matrix.fromBlocks A 0 C D).invertibleOfLeftInverse (Matrix.fromBlocks (⅟ A) 0 (-(⅟ D * C * ⅟ A)) ⅟ D) ⋯

theorem Matrix.invOf_fromBlocks_zero₂₁_eq {m : Type u_2} {n : Type u_3} {α : Type u_4} [Fintype m] [Fintype n] [DecidableEq m] [DecidableEq n] [CommRing α] (A : Matrix m m α) (B : Matrix m n α) (D : Matrix n n α) [Invertible A] [Invertible D] [Invertible (fromBlocks A B 0 D)] :

⅟ (fromBlocks A B 0 D) = fromBlocks (⅟ A) (-(⅟ A * B * ⅟ D)) 0 ⅟ D

theorem Matrix.invOf_fromBlocks_zero₁₂_eq {m : Type u_2} {n : Type u_3} {α : Type u_4} [Fintype m] [Fintype n] [DecidableEq m] [DecidableEq n] [CommRing α] (A : Matrix m m α) (C : Matrix n m α) (D : Matrix n n α) [Invertible A] [Invertible D] [Invertible (fromBlocks A 0 C D)] :

⅟ (fromBlocks A 0 C D) = fromBlocks (⅟ A) 0 (-(⅟ D * C * ⅟ A)) ⅟ D

def Matrix.invertibleOfFromBlocksZero₂₁Invertible {m : Type u_2} {n : Type u_3} {α : Type u_4} [Fintype m] [Fintype n] [DecidableEq m] [DecidableEq n] [CommRing α] (A : Matrix m m α) (B : Matrix m n α) (D : Matrix n n α) [Invertible (fromBlocks A B 0 D)] :

Invertible A × Invertible D

Both diagonal entries of an invertible upper-block-triangular matrix are invertible (by reading off the diagonal entries of the inverse).

Equations

A.invertibleOfFromBlocksZero₂₁Invertible B D = (A.invertibleOfLeftInverse (⅟ (Matrix.fromBlocks A B 0 D)).toBlocks₁₁ ⋯, D.invertibleOfRightInverse (⅟ (Matrix.fromBlocks A B 0 D)).toBlocks₂₂ ⋯)

def Matrix.invertibleOfFromBlocksZero₁₂Invertible {m : Type u_2} {n : Type u_3} {α : Type u_4} [Fintype m] [Fintype n] [DecidableEq m] [DecidableEq n] [CommRing α] (A : Matrix m m α) (C : Matrix n m α) (D : Matrix n n α) [Invertible (fromBlocks A 0 C D)] :

Invertible A × Invertible D

Both diagonal entries of an invertible lower-block-triangular matrix are invertible (by reading off the diagonal entries of the inverse).

Equations

A.invertibleOfFromBlocksZero₁₂Invertible C D = (A.invertibleOfRightInverse (⅟ (Matrix.fromBlocks A 0 C D)).toBlocks₁₁ ⋯, D.invertibleOfLeftInverse (⅟ (Matrix.fromBlocks A 0 C D)).toBlocks₂₂ ⋯)

def Matrix.fromBlocksZero₂₁InvertibleEquiv {m : Type u_2} {n : Type u_3} {α : Type u_4} [Fintype m] [Fintype n] [DecidableEq m] [DecidableEq n] [CommRing α] (A : Matrix m m α) (B : Matrix m n α) (D : Matrix n n α) :

Invertible (fromBlocks A B 0 D) ≃ Invertible A × Invertible D

invertibleOfFromBlocksZero₂₁Invertible and Matrix.fromBlocksZero₂₁Invertible form an equivalence.

Equations

One or more equations did not get rendered due to their size.

def Matrix.fromBlocksZero₁₂InvertibleEquiv {m : Type u_2} {n : Type u_3} {α : Type u_4} [Fintype m] [Fintype n] [DecidableEq m] [DecidableEq n] [CommRing α] (A : Matrix m m α) (C : Matrix n m α) (D : Matrix n n α) :

Invertible (fromBlocks A 0 C D) ≃ Invertible A × Invertible D

invertibleOfFromBlocksZero₁₂Invertible and Matrix.fromBlocksZero₁₂Invertible form an equivalence.

Equations

One or more equations did not get rendered due to their size.

@[simp]

theorem Matrix.isUnit_fromBlocks_zero₂₁ {m : Type u_2} {n : Type u_3} {α : Type u_4} [Fintype m] [Fintype n] [DecidableEq m] [DecidableEq n] [CommRing α] {A : Matrix m m α} {B : Matrix m n α} {D : Matrix n n α} :

IsUnit (fromBlocks A B 0 D) ↔ IsUnit A ∧ IsUnit D

An upper block-triangular matrix is invertible iff both elements of its diagonal are.

This is a propositional form of Matrix.fromBlocksZero₂₁InvertibleEquiv.

@[simp]

theorem Matrix.isUnit_fromBlocks_zero₁₂ {m : Type u_2} {n : Type u_3} {α : Type u_4} [Fintype m] [Fintype n] [DecidableEq m] [DecidableEq n] [CommRing α] {A : Matrix m m α} {C : Matrix n m α} {D : Matrix n n α} :

IsUnit (fromBlocks A 0 C D) ↔ IsUnit A ∧ IsUnit D

A lower block-triangular matrix is invertible iff both elements of its diagonal are.

This is a propositional form of Matrix.fromBlocksZero₁₂InvertibleEquiv forms an iff.

theorem Matrix.inv_fromBlocks_zero₂₁_of_isUnit_iff {m : Type u_2} {n : Type u_3} {α : Type u_4} [Fintype m] [Fintype n] [DecidableEq m] [DecidableEq n] [CommRing α] (A : Matrix m m α) (B : Matrix m n α) (D : Matrix n n α) (hAD : IsUnit A ↔ IsUnit D) :

(fromBlocks A B 0 D)⁻¹ = fromBlocks A⁻¹ (-(A⁻¹ * B * D⁻¹)) 0 D⁻¹

An expression for the inverse of an upper block-triangular matrix, when either both elements of diagonal are invertible, or both are not.

theorem Matrix.inv_fromBlocks_zero₁₂_of_isUnit_iff {m : Type u_2} {n : Type u_3} {α : Type u_4} [Fintype m] [Fintype n] [DecidableEq m] [DecidableEq n] [CommRing α] (A : Matrix m m α) (C : Matrix n m α) (D : Matrix n n α) (hAD : IsUnit A ↔ IsUnit D) :

(fromBlocks A 0 C D)⁻¹ = fromBlocks A⁻¹ 0 (-(D⁻¹ * C * A⁻¹)) D⁻¹

An expression for the inverse of a lower block-triangular matrix, when either both elements of diagonal are invertible, or both are not.

2×2 block matrices #

General 2×2 block matrices #

def Matrix.fromBlocks₂₂Invertible {m : Type u_2} {n : Type u_3} {α : Type u_4} [Fintype m] [Fintype n] [DecidableEq m] [DecidableEq n] [CommRing α] (A : Matrix m m α) (B : Matrix m n α) (C : Matrix n m α) (D : Matrix n n α) [Invertible D] [Invertible (A - B * ⅟ D * C)] :

Invertible (fromBlocks A B C D)

A block matrix is invertible if the bottom right corner and the corresponding schur complement is.

Equations

One or more equations did not get rendered due to their size.

def Matrix.fromBlocks₁₁Invertible {m : Type u_2} {n : Type u_3} {α : Type u_4} [Fintype m] [Fintype n] [DecidableEq m] [DecidableEq n] [CommRing α] (A : Matrix m m α) (B : Matrix m n α) (C : Matrix n m α) (D : Matrix n n α) [Invertible A] [Invertible (D - C * ⅟ A * B)] :

Invertible (fromBlocks A B C D)

A block matrix is invertible if the top left corner and the corresponding schur complement is.

Equations

One or more equations did not get rendered due to their size.

theorem Matrix.invOf_fromBlocks₂₂_eq {m : Type u_2} {n : Type u_3} {α : Type u_4} [Fintype m] [Fintype n] [DecidableEq m] [DecidableEq n] [CommRing α] (A : Matrix m m α) (B : Matrix m n α) (C : Matrix n m α) (D : Matrix n n α) [Invertible D] [Invertible (A - B * ⅟ D * C)] [Invertible (fromBlocks A B C D)] :

⅟ (fromBlocks A B C D) = fromBlocks (⅟ (A - B * ⅟ D * C)) (-(⅟ (A - B * ⅟ D * C) * B * ⅟ D)) (-(⅟ D * C * ⅟ (A - B * ⅟ D * C))) (⅟ D + ⅟ D * C * ⅟ (A - B * ⅟ D * C) * B * ⅟ D)

theorem Matrix.invOf_fromBlocks₁₁_eq {m : Type u_2} {n : Type u_3} {α : Type u_4} [Fintype m] [Fintype n] [DecidableEq m] [DecidableEq n] [CommRing α] (A : Matrix m m α) (B : Matrix m n α) (C : Matrix n m α) (D : Matrix n n α) [Invertible A] [Invertible (D - C * ⅟ A * B)] [Invertible (fromBlocks A B C D)] :

⅟ (fromBlocks A B C D) = fromBlocks (⅟ A + ⅟ A * B * ⅟ (D - C * ⅟ A * B) * C * ⅟ A) (-(⅟ A * B * ⅟ (D - C * ⅟ A * B))) (-(⅟ (D - C * ⅟ A * B) * C * ⅟ A)) ⅟ (D - C * ⅟ A * B)

def Matrix.invertibleOfFromBlocks₂₂Invertible {m : Type u_2} {n : Type u_3} {α : Type u_4} [Fintype m] [Fintype n] [DecidableEq m] [DecidableEq n] [CommRing α] (A : Matrix m m α) (B : Matrix m n α) (C : Matrix n m α) (D : Matrix n n α) [Invertible D] [Invertible (fromBlocks A B C D)] :

Invertible (A - B * ⅟ D * C)

If a block matrix is invertible and so is its bottom left element, then so is the corresponding Schur complement.

Equations

A.invertibleOfFromBlocks₂₂Invertible B C D = ((A - B * ⅟ D * C).invertibleOfFromBlocksZero₁₂Invertible 0 D).1

def Matrix.invertibleOfFromBlocks₁₁Invertible {m : Type u_2} {n : Type u_3} {α : Type u_4} [Fintype m] [Fintype n] [DecidableEq m] [DecidableEq n] [CommRing α] (A : Matrix m m α) (B : Matrix m n α) (C : Matrix n m α) (D : Matrix n n α) [Invertible A] [Invertible (fromBlocks A B C D)] :

Invertible (D - C * ⅟ A * B)

If a block matrix is invertible and so is its bottom left element, then so is the corresponding Schur complement.

Equations

A.invertibleOfFromBlocks₁₁Invertible B C D = D.invertibleOfFromBlocks₂₂Invertible C B A

def Matrix.invertibleEquivFromBlocks₂₂Invertible {m : Type u_2} {n : Type u_3} {α : Type u_4} [Fintype m] [Fintype n] [DecidableEq m] [DecidableEq n] [CommRing α] (A : Matrix m m α) (B : Matrix m n α) (C : Matrix n m α) (D : Matrix n n α) [Invertible D] :

Invertible (fromBlocks A B C D) ≃ Invertible (A - B * ⅟ D * C)

Matrix.invertibleOfFromBlocks₂₂Invertible and Matrix.fromBlocks₂₂Invertible as an equivalence.

Equations

One or more equations did not get rendered due to their size.

def Matrix.invertibleEquivFromBlocks₁₁Invertible {m : Type u_2} {n : Type u_3} {α : Type u_4} [Fintype m] [Fintype n] [DecidableEq m] [DecidableEq n] [CommRing α] (A : Matrix m m α) (B : Matrix m n α) (C : Matrix n m α) (D : Matrix n n α) [Invertible A] :

Invertible (fromBlocks A B C D) ≃ Invertible (D - C * ⅟ A * B)

Matrix.invertibleOfFromBlocks₁₁Invertible and Matrix.fromBlocks₁₁Invertible as an equivalence.

Equations

One or more equations did not get rendered due to their size.

theorem Matrix.isUnit_fromBlocks_iff_of_invertible₂₂ {m : Type u_2} {n : Type u_3} {α : Type u_4} [Fintype m] [Fintype n] [DecidableEq m] [DecidableEq n] [CommRing α] {A : Matrix m m α} {B : Matrix m n α} {C : Matrix n m α} {D : Matrix n n α} [Invertible D] :

IsUnit (fromBlocks A B C D) ↔ IsUnit (A - B * ⅟ D * C)

If the bottom-left element of a block matrix is invertible, then the whole matrix is invertible iff the corresponding schur complement is.

theorem Matrix.isUnit_fromBlocks_iff_of_invertible₁₁ {m : Type u_2} {n : Type u_3} {α : Type u_4} [Fintype m] [Fintype n] [DecidableEq m] [DecidableEq n] [CommRing α] {A : Matrix m m α} {B : Matrix m n α} {C : Matrix n m α} {D : Matrix n n α} [Invertible A] :

IsUnit (fromBlocks A B C D) ↔ IsUnit (D - C * ⅟ A * B)

If the top-right element of a block matrix is invertible, then the whole matrix is invertible iff the corresponding schur complement is.

Lemmas about `Matrix.det` #

theorem Matrix.det_fromBlocks₁₁ {m : Type u_2} {n : Type u_3} {α : Type u_4} [Fintype m] [Fintype n] [DecidableEq m] [DecidableEq n] [CommRing α] (A : Matrix m m α) (B : Matrix m n α) (C : Matrix n m α) (D : Matrix n n α) [Invertible A] :

(fromBlocks A B C D).det = A.det * (D - C * ⅟ A * B).det

Determinant of a 2×2 block matrix, expanded around an invertible top left element in terms of the Schur complement.

@[simp]

theorem Matrix.det_fromBlocks_one₁₁ {m : Type u_2} {n : Type u_3} {α : Type u_4} [Fintype m] [Fintype n] [DecidableEq m] [DecidableEq n] [CommRing α] (B : Matrix m n α) (C : Matrix n m α) (D : Matrix n n α) :

(fromBlocks 1 B C D).det = (D - C * B).det

theorem Matrix.det_fromBlocks₂₂ {m : Type u_2} {n : Type u_3} {α : Type u_4} [Fintype m] [Fintype n] [DecidableEq m] [DecidableEq n] [CommRing α] (A : Matrix m m α) (B : Matrix m n α) (C : Matrix n m α) (D : Matrix n n α) [Invertible D] :

(fromBlocks A B C D).det = D.det * (A - B * ⅟ D * C).det

Determinant of a 2×2 block matrix, expanded around an invertible bottom right element in terms of the Schur complement.

@[simp]

theorem Matrix.det_fromBlocks_one₂₂ {m : Type u_2} {n : Type u_3} {α : Type u_4} [Fintype m] [Fintype n] [DecidableEq m] [DecidableEq n] [CommRing α] (A : Matrix m m α) (B : Matrix m n α) (C : Matrix n m α) :

(fromBlocks A B C 1).det = (A - B * C).det

theorem Matrix.det_one_add_mul_comm {m : Type u_2} {n : Type u_3} {α : Type u_4} [Fintype m] [Fintype n] [DecidableEq m] [DecidableEq n] [CommRing α] (A : Matrix m n α) (B : Matrix n m α) :

(1 + A * B).det = (1 + B * A).det

The Weinstein–Aronszajn identity. Note the 1 on the LHS is of shape m×m, while the 1 on the RHS is of shape n×n.

theorem Matrix.det_mul_add_one_comm {m : Type u_2} {n : Type u_3} {α : Type u_4} [Fintype m] [Fintype n] [DecidableEq m] [DecidableEq n] [CommRing α] (A : Matrix m n α) (B : Matrix n m α) :

(A * B + 1).det = (B * A + 1).det

Alternate statement of the Weinstein–Aronszajn identity

theorem Matrix.det_one_sub_mul_comm {m : Type u_2} {n : Type u_3} {α : Type u_4} [Fintype m] [Fintype n] [DecidableEq m] [DecidableEq n] [CommRing α] (A : Matrix m n α) (B : Matrix n m α) :

(1 - A * B).det = (1 - B * A).det

theorem Matrix.det_one_add_replicateCol_mul_replicateRow {m : Type u_2} {α : Type u_4} [Fintype m] [DecidableEq m] [CommRing α] {ι : Type u_5} [Unique ι] (u v : m → α) :

(1 + replicateCol ι u * replicateRow ι v).det = 1 + v ⬝ᵥ u

A special case of the Matrix determinant lemma for when A = I.

@[deprecated Matrix.det_one_add_replicateCol_mul_replicateRow (since := "2025-03-20")]

theorem Matrix.det_one_add_col_mul_row {m : Type u_2} {α : Type u_4} [Fintype m] [DecidableEq m] [CommRing α] {ι : Type u_5} [Unique ι] (u v : m → α) :

(1 + replicateCol ι u * replicateRow ι v).det = 1 + v ⬝ᵥ u

Alias of Matrix.det_one_add_replicateCol_mul_replicateRow.

A special case of the Matrix determinant lemma for when A = I.

theorem Matrix.det_add_replicateCol_mul_replicateRow {m : Type u_2} {α : Type u_4} [Fintype m] [DecidableEq m] [CommRing α] {ι : Type u_5} [Unique ι] {A : Matrix m m α} (hA : IsUnit A.det) (u v : m → α) :

(A + replicateCol ι u * replicateRow ι v).det = A.det * (1 + replicateRow ι v * A⁻¹ * replicateCol ι u).det

The Matrix determinant lemma

TODO: show the more general version without hA : IsUnit A.det as (A + replicateCol u * replicateRow v).det = A.det + v ⬝ᵥ (adjugate A) *ᵥ u.

@[deprecated Matrix.det_add_replicateCol_mul_replicateRow (since := "2025-03-20")]

theorem Matrix.det_add_col_mul_row {m : Type u_2} {α : Type u_4} [Fintype m] [DecidableEq m] [CommRing α] {ι : Type u_5} [Unique ι] {A : Matrix m m α} (hA : IsUnit A.det) (u v : m → α) :

(A + replicateCol ι u * replicateRow ι v).det = A.det * (1 + replicateRow ι v * A⁻¹ * replicateCol ι u).det

Alias of Matrix.det_add_replicateCol_mul_replicateRow.

The Matrix determinant lemma

TODO: show the more general version without hA : IsUnit A.det as (A + replicateCol u * replicateRow v).det = A.det + v ⬝ᵥ (adjugate A) *ᵥ u.

theorem Matrix.det_add_mul {m : Type u_2} {n : Type u_3} {α : Type u_4} [Fintype m] [Fintype n] [DecidableEq m] [DecidableEq n] [CommRing α] {A : Matrix m m α} (U : Matrix m n α) (V : Matrix n m α) (hA : IsUnit A.det) :

(A + U * V).det = A.det * (1 + V * A⁻¹ * U).det

A generalization of the Matrix determinant lemma

Lemmas about `ℝ` and `ℂ` and other `StarOrderedRing`s #

def Matrix.«term_⊕ᵥ_» :

Lean.TrailingParserDescr

Notation for Sum.elim, scoped within the Matrix namespace.

Equations

Matrix.«term_⊕ᵥ_» = Lean.ParserDescr.trailingNode `Matrix.«term_⊕ᵥ_» 65 65 (Lean.ParserDescr.binary `andthen (Lean.ParserDescr.symbol " ⊕ᵥ ") (Lean.ParserDescr.cat `term 66))

theorem Matrix.schur_complement_eq₁₁ {m : Type u_2} {n : Type u_3} {𝕜 : Type u_5} [CommRing 𝕜] [StarRing 𝕜] [Fintype m] [DecidableEq m] [Fintype n] {A : Matrix m m 𝕜} (B : Matrix m n 𝕜) (D : Matrix n n 𝕜) (x : m → 𝕜) (y : n → 𝕜) [Invertible A] (hA : A.IsHermitian) :

vecMul (star (Sum.elim x y)) (fromBlocks A B B.conjTranspose D) ⬝ᵥ Sum.elim x y = vecMul (star (x + (A⁻¹ * B).mulVec y)) A ⬝ᵥ (x + (A⁻¹ * B).mulVec y) + vecMul (star y) (D - B.conjTranspose * A⁻¹ * B) ⬝ᵥ y

theorem Matrix.schur_complement_eq₂₂ {m : Type u_2} {n : Type u_3} {𝕜 : Type u_5} [CommRing 𝕜] [StarRing 𝕜] [Fintype m] [Fintype n] [DecidableEq n] (A : Matrix m m 𝕜) (B : Matrix m n 𝕜) {D : Matrix n n 𝕜} (x : m → 𝕜) (y : n → 𝕜) [Invertible D] (hD : D.IsHermitian) :

vecMul (star (Sum.elim x y)) (fromBlocks A B B.conjTranspose D) ⬝ᵥ Sum.elim x y = vecMul (star ((D⁻¹ * B.conjTranspose).mulVec x + y)) D ⬝ᵥ ((D⁻¹ * B.conjTranspose).mulVec x + y) + vecMul (star x) (A - B * D⁻¹ * B.conjTranspose) ⬝ᵥ x

theorem Matrix.IsHermitian.fromBlocks₁₁ {m : Type u_2} {n : Type u_3} {𝕜 : Type u_5} [CommRing 𝕜] [StarRing 𝕜] [Fintype m] [DecidableEq m] {A : Matrix m m 𝕜} (B : Matrix m n 𝕜) (D : Matrix n n 𝕜) (hA : A.IsHermitian) :

(Matrix.fromBlocks A B B.conjTranspose D).IsHermitian ↔ (D - B.conjTranspose * A⁻¹ * B).IsHermitian

theorem Matrix.IsHermitian.fromBlocks₂₂ {m : Type u_2} {n : Type u_3} {𝕜 : Type u_5} [CommRing 𝕜] [StarRing 𝕜] [Fintype n] [DecidableEq n] (A : Matrix m m 𝕜) (B : Matrix m n 𝕜) {D : Matrix n n 𝕜} (hD : D.IsHermitian) :

(Matrix.fromBlocks A B B.conjTranspose D).IsHermitian ↔ (A - B * D⁻¹ * B.conjTranspose).IsHermitian

theorem Matrix.PosSemidef.fromBlocks₁₁ {m : Type u_2} {n : Type u_3} {𝕜 : Type u_5} [CommRing 𝕜] [StarRing 𝕜] [PartialOrder 𝕜] [StarOrderedRing 𝕜] [Fintype m] [DecidableEq m] [Fintype n] {A : Matrix m m 𝕜} (B : Matrix m n 𝕜) (D : Matrix n n 𝕜) (hA : A.PosDef) [Invertible A] :

(fromBlocks A B B.conjTranspose D).PosSemidef ↔ (D - B.conjTranspose * A⁻¹ * B).PosSemidef

theorem Matrix.PosSemidef.fromBlocks₂₂ {m : Type u_2} {n : Type u_3} {𝕜 : Type u_5} [CommRing 𝕜] [StarRing 𝕜] [PartialOrder 𝕜] [StarOrderedRing 𝕜] [Fintype m] [Fintype n] [DecidableEq n] (A : Matrix m m 𝕜) (B : Matrix m n 𝕜) {D : Matrix n n 𝕜} (hD : D.PosDef) [Invertible D] :

(fromBlocks A B B.conjTranspose D).PosSemidef ↔ (A - B * D⁻¹ * B.conjTranspose).PosSemidef