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Mathlib.Analysis.NormedSpace.MatrixExponential

Lemmas about the matrix exponential #

In this file, we provide results about exp on Matrixs over a topological or normed algebra. Note that generic results over all topological spaces such as exp_zero can be used on matrices without issue, so are not repeated here. The topological results specific to matrices are:

Lemmas like exp_add_of_commute require a canonical norm on the type; while there are multiple sensible choices for the norm of a Matrix (Matrix.normedAddCommGroup, Matrix.frobeniusNormedAddCommGroup, Matrix.linftyOpNormedAddCommGroup), none of them are canonical. In an application where a particular norm is chosen using local attribute [instance], then the usual lemmas about exp are fine. When choosing a norm is undesirable, the results in this file can be used.

In this file, we copy across the lemmas about exp, but hide the requirement for a norm inside the proof.

Additionally, we prove some results about matrix.has_inv and matrix.div_inv_monoid, as the results for general rings are instead stated about Ring.inverse:

TODO #

Show that Matrix.det (exp 𝕂 A) = exp 𝕂 (Matrix.trace A)

References #

https://en.wikipedia.org/wiki/Matrix_exponential

theorem Matrix.exp_diagonal (𝕂 : Type u_1) {m : Type u_2} {𝔸 : Type u_6} [Fintype m] [DecidableEq m] [Field 𝕂] [Ring 𝔸] [TopologicalSpace 𝔸] [TopologicalRing 𝔸] [Algebra 𝕂 𝔸] [T2Space 𝔸] (v : m → 𝔸) :

exp 𝕂 (Matrix.diagonal v) = Matrix.diagonal (exp 𝕂 v)

theorem Matrix.exp_blockDiagonal (𝕂 : Type u_1) {m : Type u_2} {n : Type u_3} {𝔸 : Type u_6} [Fintype m] [DecidableEq m] [Fintype n] [DecidableEq n] [Field 𝕂] [Ring 𝔸] [TopologicalSpace 𝔸] [TopologicalRing 𝔸] [Algebra 𝕂 𝔸] [T2Space 𝔸] (v : m → Matrix n n 𝔸) :

exp 𝕂 (Matrix.blockDiagonal v) = Matrix.blockDiagonal (exp 𝕂 v)

theorem Matrix.exp_blockDiagonal' (𝕂 : Type u_1) {m : Type u_2} {n' : m → Type u_5} {𝔸 : Type u_6} [Fintype m] [DecidableEq m] [(i : m) → Fintype (n' i)] [(i : m) → DecidableEq (n' i)] [Field 𝕂] [Ring 𝔸] [TopologicalSpace 𝔸] [TopologicalRing 𝔸] [Algebra 𝕂 𝔸] [T2Space 𝔸] (v : (i : m) → Matrix (n' i) (n' i) 𝔸) :

exp 𝕂 (Matrix.blockDiagonal' v) = Matrix.blockDiagonal' (exp 𝕂 v)

theorem Matrix.exp_conjTranspose (𝕂 : Type u_1) {m : Type u_2} {𝔸 : Type u_6} [Fintype m] [DecidableEq m] [Field 𝕂] [Ring 𝔸] [TopologicalSpace 𝔸] [TopologicalRing 𝔸] [Algebra 𝕂 𝔸] [T2Space 𝔸] [StarRing 𝔸] [ContinuousStar 𝔸] (A : Matrix m m 𝔸) :

exp 𝕂 (Matrix.conjTranspose A) = Matrix.conjTranspose (exp 𝕂 A)

theorem Matrix.IsHermitian.exp (𝕂 : Type u_1) {m : Type u_2} {𝔸 : Type u_6} [Fintype m] [DecidableEq m] [Field 𝕂] [Ring 𝔸] [TopologicalSpace 𝔸] [TopologicalRing 𝔸] [Algebra 𝕂 𝔸] [T2Space 𝔸] [StarRing 𝔸] [ContinuousStar 𝔸] {A : Matrix m m 𝔸} (h : Matrix.IsHermitian A) :

Matrix.IsHermitian (exp 𝕂 A)

theorem Matrix.exp_transpose (𝕂 : Type u_1) {m : Type u_2} {𝔸 : Type u_6} [Fintype m] [DecidableEq m] [Field 𝕂] [CommRing 𝔸] [TopologicalSpace 𝔸] [TopologicalRing 𝔸] [Algebra 𝕂 𝔸] [T2Space 𝔸] (A : Matrix m m 𝔸) :

exp 𝕂 (Matrix.transpose A) = Matrix.transpose (exp 𝕂 A)

theorem Matrix.IsSymm.exp (𝕂 : Type u_1) {m : Type u_2} {𝔸 : Type u_6} [Fintype m] [DecidableEq m] [Field 𝕂] [CommRing 𝔸] [TopologicalSpace 𝔸] [TopologicalRing 𝔸] [Algebra 𝕂 𝔸] [T2Space 𝔸] {A : Matrix m m 𝔸} (h : Matrix.IsSymm A) :

Matrix.IsSymm (exp 𝕂 A)

theorem Matrix.exp_add_of_commute (𝕂 : Type u_1) {m : Type u_2} {𝔸 : Type u_6} [IsROrC 𝕂] [Fintype m] [DecidableEq m] [NormedRing 𝔸] [NormedAlgebra 𝕂 𝔸] [CompleteSpace 𝔸] (A : Matrix m m 𝔸) (B : Matrix m m 𝔸) (h : Commute A B) :

exp 𝕂 (A + B) = exp 𝕂 A * exp 𝕂 B

theorem Matrix.exp_sum_of_commute (𝕂 : Type u_1) {m : Type u_2} {𝔸 : Type u_6} [IsROrC 𝕂] [Fintype m] [DecidableEq m] [NormedRing 𝔸] [NormedAlgebra 𝕂 𝔸] [CompleteSpace 𝔸] {ι : Type u_7} (s : Finset ι) (f : ι → Matrix m m 𝔸) (h : Set.Pairwise ↑s fun i j => Commute (f i) (f j)) :

exp 𝕂 (Finset.sum s fun i => f i) = Finset.noncommProd s (fun i => exp 𝕂 (f i)) (_ : ∀ (i : ι), i ∈ ↑s → ∀ (j : ι), j ∈ ↑s → i ≠ j → Commute (exp 𝕂 (f i)) (exp 𝕂 (f j)))

theorem Matrix.exp_nsmul (𝕂 : Type u_1) {m : Type u_2} {𝔸 : Type u_6} [IsROrC 𝕂] [Fintype m] [DecidableEq m] [NormedRing 𝔸] [NormedAlgebra 𝕂 𝔸] [CompleteSpace 𝔸] (n : ℕ) (A : Matrix m m 𝔸) :

exp 𝕂 (n • A) = exp 𝕂 A ^ n

theorem Matrix.isUnit_exp (𝕂 : Type u_1) {m : Type u_2} {𝔸 : Type u_6} [IsROrC 𝕂] [Fintype m] [DecidableEq m] [NormedRing 𝔸] [NormedAlgebra 𝕂 𝔸] [CompleteSpace 𝔸] (A : Matrix m m 𝔸) :

IsUnit (exp 𝕂 A)

theorem Matrix.exp_units_conj (𝕂 : Type u_1) {m : Type u_2} {𝔸 : Type u_6} [IsROrC 𝕂] [Fintype m] [DecidableEq m] [NormedRing 𝔸] [NormedAlgebra 𝕂 𝔸] [CompleteSpace 𝔸] (U : (Matrix m m 𝔸)ˣ) (A : Matrix m m 𝔸) :

exp 𝕂 (↑U * A * ↑U⁻¹) = ↑U * exp 𝕂 A * ↑U⁻¹

theorem Matrix.exp_units_conj' (𝕂 : Type u_1) {m : Type u_2} {𝔸 : Type u_6} [IsROrC 𝕂] [Fintype m] [DecidableEq m] [NormedRing 𝔸] [NormedAlgebra 𝕂 𝔸] [CompleteSpace 𝔸] (U : (Matrix m m 𝔸)ˣ) (A : Matrix m m 𝔸) :

exp 𝕂 (↑U⁻¹ * A * ↑U) = ↑U⁻¹ * exp 𝕂 A * ↑U

theorem Matrix.exp_neg (𝕂 : Type u_1) {m : Type u_2} {𝔸 : Type u_6} [IsROrC 𝕂] [Fintype m] [DecidableEq m] [NormedCommRing 𝔸] [NormedAlgebra 𝕂 𝔸] [CompleteSpace 𝔸] (A : Matrix m m 𝔸) :

exp 𝕂 (-A) = (exp 𝕂 A)⁻¹

theorem Matrix.exp_zsmul (𝕂 : Type u_1) {m : Type u_2} {𝔸 : Type u_6} [IsROrC 𝕂] [Fintype m] [DecidableEq m] [NormedCommRing 𝔸] [NormedAlgebra 𝕂 𝔸] [CompleteSpace 𝔸] (z : ℤ) (A : Matrix m m 𝔸) :

exp 𝕂 (z • A) = exp 𝕂 A ^ z

theorem Matrix.exp_conj (𝕂 : Type u_1) {m : Type u_2} {𝔸 : Type u_6} [IsROrC 𝕂] [Fintype m] [DecidableEq m] [NormedCommRing 𝔸] [NormedAlgebra 𝕂 𝔸] [CompleteSpace 𝔸] (U : Matrix m m 𝔸) (A : Matrix m m 𝔸) (hy : IsUnit U) :

exp 𝕂 (U * A * U⁻¹) = U * exp 𝕂 A * U⁻¹

theorem Matrix.exp_conj' (𝕂 : Type u_1) {m : Type u_2} {𝔸 : Type u_6} [IsROrC 𝕂] [Fintype m] [DecidableEq m] [NormedCommRing 𝔸] [NormedAlgebra 𝕂 𝔸] [CompleteSpace 𝔸] (U : Matrix m m 𝔸) (A : Matrix m m 𝔸) (hy : IsUnit U) :

exp 𝕂 (U⁻¹ * A * U) = U⁻¹ * exp 𝕂 A * U