Hermitian matrices #

This file defines hermitian matrices and some basic results about them.

See also IsSelfAdjoint, which generalizes this definition to other star rings.

Main definition #

Matrix.IsHermitian : a matrix A : Matrix n n α is hermitian if Aᴴ = A.

Tags #

self-adjoint matrix, hermitian matrix

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def Matrix.IsHermitian {α : Type u_1} {n : Type u_4} [Star α] (A : Matrix n n α) :

Prop

A matrix is hermitian if it is equal to its conjugate transpose. On the reals, this definition captures symmetric matrices.

Equations

A.IsHermitian = (A.conjTranspose = A)

Instances For

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instance Matrix.instDecidableIsHermitianOfEqConjTranspose {α : Type u_1} {n : Type u_4} [Star α] (A : Matrix n n α) [Decidable (A.conjTranspose = A)] :

Decidable A.IsHermitian

Equations

A.instDecidableIsHermitianOfEqConjTranspose = inferInstanceAs (Decidable (A.conjTranspose = A))

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theorem Matrix.IsHermitian.eq {α : Type u_1} {n : Type u_4} [Star α] {A : Matrix n n α} (h : A.IsHermitian) :

A.conjTranspose = A

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theorem Matrix.IsHermitian.isSelfAdjoint {α : Type u_1} {n : Type u_4} [Star α] {A : Matrix n n α} (h : A.IsHermitian) :

IsSelfAdjoint A

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theorem Matrix.IsHermitian.ext {α : Type u_1} {n : Type u_4} [Star α] {A : Matrix n n α} :

(∀ (i j : n), star (A j i) = A i j) → A.IsHermitian

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theorem Matrix.IsHermitian.apply {α : Type u_1} {n : Type u_4} [Star α] {A : Matrix n n α} (h : A.IsHermitian) (i j : n) :

star (A j i) = A i j

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theorem Matrix.IsHermitian.ext_iff {α : Type u_1} {n : Type u_4} [Star α] {A : Matrix n n α} :

A.IsHermitian ↔ ∀ (i j : n), star (A j i) = A i j

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@[simp]

theorem Matrix.IsHermitian.map {α : Type u_1} {β : Type u_2} {n : Type u_4} [Star α] [Star β] {A : Matrix n n α} (h : A.IsHermitian) (f : α → β) (hf : Function.Semiconj f star star) :

(A.map f).IsHermitian

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theorem Matrix.IsHermitian.transpose {α : Type u_1} {n : Type u_4} [Star α] {A : Matrix n n α} (h : A.IsHermitian) :

A.transpose.IsHermitian

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@[simp]

theorem Matrix.isHermitian_transpose_iff {α : Type u_1} {n : Type u_4} [Star α] (A : Matrix n n α) :

A.transpose.IsHermitian ↔ A.IsHermitian

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theorem Matrix.IsHermitian.conjTranspose {α : Type u_1} {n : Type u_4} [Star α] {A : Matrix n n α} (h : A.IsHermitian) :

A.conjTranspose.IsHermitian

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@[simp]

theorem Matrix.IsHermitian.submatrix {α : Type u_1} {m : Type u_3} {n : Type u_4} [Star α] {A : Matrix n n α} (h : A.IsHermitian) (f : m → n) :

(A.submatrix f f).IsHermitian

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@[simp]

theorem Matrix.isHermitian_submatrix_equiv {α : Type u_1} {m : Type u_3} {n : Type u_4} [Star α] {A : Matrix n n α} (e : m ≃ n) :

(A.submatrix ⇑e ⇑e).IsHermitian ↔ A.IsHermitian

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@[simp]

theorem Matrix.isHermitian_conjTranspose_iff {α : Type u_1} {n : Type u_4} [InvolutiveStar α] (A : Matrix n n α) :

A.conjTranspose.IsHermitian ↔ A.IsHermitian

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theorem Matrix.IsHermitian.fromBlocks {α : Type u_1} {m : Type u_3} {n : Type u_4} [InvolutiveStar α] {A : Matrix m m α} {B : Matrix m n α} {C : Matrix n m α} {D : Matrix n n α} (hA : A.IsHermitian) (hBC : B.conjTranspose = C) (hD : D.IsHermitian) :

(Matrix.fromBlocks A B C D).IsHermitian

A block matrix A.from_blocks B C D is hermitian, if A and D are hermitian and Bᴴ = C.

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theorem Matrix.isHermitian_fromBlocks_iff {α : Type u_1} {m : Type u_3} {n : Type u_4} [InvolutiveStar α] {A : Matrix m m α} {B : Matrix m n α} {C : Matrix n m α} {D : Matrix n n α} :

(fromBlocks A B C D).IsHermitian ↔ A.IsHermitian ∧ B.conjTranspose = C ∧ C.conjTranspose = B ∧ D.IsHermitian

This is the iff version of Matrix.IsHermitian.fromBlocks.

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theorem Matrix.isHermitian_diagonal_of_self_adjoint {α : Type u_1} {n : Type u_4} [AddMonoid α] [StarAddMonoid α] [DecidableEq n] (v : n → α) (h : IsSelfAdjoint v) :

(diagonal v).IsHermitian

A diagonal matrix is hermitian if the entries are self-adjoint (as a vector)

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theorem Matrix.isHermitian_diagonal_iff {α : Type u_1} {n : Type u_4} [AddMonoid α] [StarAddMonoid α] [DecidableEq n] {d : n → α} :

(diagonal d).IsHermitian ↔ ∀ (i : n), IsSelfAdjoint (d i)

A diagonal matrix is hermitian if each diagonal entry is self-adjoint

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@[simp]

theorem Matrix.isHermitian_diagonal {α : Type u_1} {n : Type u_4} [AddMonoid α] [StarAddMonoid α] [TrivialStar α] [DecidableEq n] (v : n → α) :

(diagonal v).IsHermitian

A diagonal matrix is hermitian if the entries have the trivial star operation (such as on the reals).

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@[simp]

theorem Matrix.isHermitian_zero {α : Type u_1} {n : Type u_4} [AddMonoid α] [StarAddMonoid α] :

IsHermitian 0

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@[simp]

theorem Matrix.IsHermitian.add {α : Type u_1} {n : Type u_4} [AddMonoid α] [StarAddMonoid α] {A B : Matrix n n α} (hA : A.IsHermitian) (hB : B.IsHermitian) :

(A + B).IsHermitian

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theorem Matrix.isHermitian_add_transpose_self {α : Type u_1} {n : Type u_4} [AddCommMonoid α] [StarAddMonoid α] (A : Matrix n n α) :

(A + A.conjTranspose).IsHermitian

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theorem Matrix.isHermitian_transpose_add_self {α : Type u_1} {n : Type u_4} [AddCommMonoid α] [StarAddMonoid α] (A : Matrix n n α) :

(A.conjTranspose + A).IsHermitian

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@[simp]

theorem Matrix.IsHermitian.neg {α : Type u_1} {n : Type u_4} [AddGroup α] [StarAddMonoid α] {A : Matrix n n α} (h : A.IsHermitian) :

(-A).IsHermitian

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@[simp]

theorem Matrix.IsHermitian.sub {α : Type u_1} {n : Type u_4} [AddGroup α] [StarAddMonoid α] {A B : Matrix n n α} (hA : A.IsHermitian) (hB : B.IsHermitian) :

(A - B).IsHermitian

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theorem Matrix.isHermitian_mul_conjTranspose_self {α : Type u_1} {m : Type u_3} {n : Type u_4} [NonUnitalSemiring α] [StarRing α] [Fintype n] (A : Matrix m n α) :

(A * A.conjTranspose).IsHermitian

Note this is more general than IsSelfAdjoint.mul_star_self as B can be rectangular.

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theorem Matrix.isHermitian_transpose_mul_self {α : Type u_1} {m : Type u_3} {n : Type u_4} [NonUnitalSemiring α] [StarRing α] [Fintype m] (A : Matrix m n α) :

(A.conjTranspose * A).IsHermitian

Note this is more general than IsSelfAdjoint.star_mul_self as B can be rectangular.

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theorem Matrix.isHermitian_conjTranspose_mul_mul {α : Type u_1} {m : Type u_3} {n : Type u_4} [NonUnitalSemiring α] [StarRing α] [Fintype m] {A : Matrix m m α} (B : Matrix m n α) (hA : A.IsHermitian) :

(B.conjTranspose * A * B).IsHermitian

Note this is more general than IsSelfAdjoint.conjugate' as B can be rectangular.

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theorem Matrix.isHermitian_mul_mul_conjTranspose {α : Type u_1} {m : Type u_3} {n : Type u_4} [NonUnitalSemiring α] [StarRing α] [Fintype m] {A : Matrix m m α} (B : Matrix n m α) (hA : A.IsHermitian) :

(B * A * B.conjTranspose).IsHermitian

Note this is more general than IsSelfAdjoint.conjugate as B can be rectangular.

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theorem Matrix.IsHermitian.commute_iff {α : Type u_1} {n : Type u_4} [NonUnitalSemiring α] [StarRing α] [Fintype n] {A B : Matrix n n α} (hA : A.IsHermitian) (hB : B.IsHermitian) :

Commute A B ↔ (A * B).IsHermitian

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@[deprecated Matrix.IsHermitian.commute_iff (since := "13-08-2025")]

theorem Matrix.commute_iff {α : Type u_1} {n : Type u_4} [NonUnitalSemiring α] [StarRing α] [Fintype n] {A B : Matrix n n α} (hA : A.IsHermitian) (hB : B.IsHermitian) :

Commute A B ↔ (A * B).IsHermitian

Alias of Matrix.IsHermitian.commute_iff.

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@[simp]

theorem Matrix.isHermitian_one {α : Type u_1} {n : Type u_4} [Semiring α] [StarRing α] [DecidableEq n] :

IsHermitian 1

Note this is more general for matrices than isSelfAdjoint_one as it does not require Fintype n, which is necessary for Monoid (Matrix n n R).

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@[simp]

theorem Matrix.isHermitian_natCast {α : Type u_1} {n : Type u_4} [Semiring α] [StarRing α] [DecidableEq n] (d : ℕ) :

(↑d).IsHermitian

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theorem Matrix.IsHermitian.pow {α : Type u_1} {n : Type u_4} [Semiring α] [StarRing α] [Fintype n] [DecidableEq n] {A : Matrix n n α} (h : A.IsHermitian) (k : ℕ) :

(A ^ k).IsHermitian

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@[simp]

theorem Matrix.isHermitian_intCast {α : Type u_1} {n : Type u_4} [Ring α] [StarRing α] [DecidableEq n] (d : ℤ) :

(↑d).IsHermitian

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theorem Matrix.IsHermitian.inv {α : Type u_1} {m : Type u_3} [CommRing α] [StarRing α] [Fintype m] [DecidableEq m] {A : Matrix m m α} (hA : A.IsHermitian) :

A⁻¹.IsHermitian

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@[simp]

theorem Matrix.isHermitian_inv {α : Type u_1} {m : Type u_3} [CommRing α] [StarRing α] [Fintype m] [DecidableEq m] (A : Matrix m m α) [Invertible A] :

A⁻¹.IsHermitian ↔ A.IsHermitian

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theorem Matrix.IsHermitian.adjugate {α : Type u_1} {m : Type u_3} [CommRing α] [StarRing α] [Fintype m] [DecidableEq m] {A : Matrix m m α} (hA : A.IsHermitian) :

A.adjugate.IsHermitian

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theorem Matrix.IsHermitian.zpow {α : Type u_1} {m : Type u_3} [CommRing α] [StarRing α] [Fintype m] [DecidableEq m] {A : Matrix m m α} (h : A.IsHermitian) (k : ℤ) :

(A ^ k).IsHermitian

Note that IsSelfAdjoint.zpow does not apply to matrices as they are not a division ring.

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theorem Matrix.IsHermitian.coe_re_apply_self {α : Type u_1} {n : Type u_4} [RCLike α] {A : Matrix n n α} (h : A.IsHermitian) (i : n) :

↑(RCLike.re (A i i)) = A i i

The diagonal elements of a complex hermitian matrix are real.

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theorem Matrix.IsHermitian.coe_re_diag {α : Type u_1} {n : Type u_4} [RCLike α] {A : Matrix n n α} (h : A.IsHermitian) :

(fun (i : n) => ↑(RCLike.re (A.diag i))) = A.diag

The diagonal elements of a complex hermitian matrix are real.

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theorem Matrix.isHermitian_iff_isSymmetric {α : Type u_1} {n : Type u_4} [RCLike α] [Fintype n] [DecidableEq n] {A : Matrix n n α} :

A.IsHermitian ↔ (toEuclideanLin A).IsSymmetric

A matrix is hermitian iff the corresponding linear map is self adjoint.

Documentation

Mathlib.LinearAlgebra.Matrix.Hermitian

Hermitian matrices #

Main definition #

Tags #