# Integer powers of square matrices #

In this file, we define integer power of matrices, relying on the nonsingular inverse definition for negative powers.

## Implementation details #

The main definition is a direct recursive call on the integer inductive type, as provided by the DivInvMonoid.Pow default implementation. The lemma names are taken from Algebra.GroupWithZero.Power.

## Tags #

matrix inverse, matrix powers

noncomputable instance Matrix.instDivInvMonoid {n' : Type u_1} [] [Fintype n'] {R : Type u_2} [] :
Equations
• Matrix.instDivInvMonoid = let __src := let_fun this := inferInstance; this; let __src_1 := let_fun this := inferInstance; this; DivInvMonoid.mk zpowRec
@[simp]
theorem Matrix.inv_pow' {n' : Type u_1} [] [Fintype n'] {R : Type u_2} [] (A : Matrix n' n' R) (n : ) :
A⁻¹ ^ n = (A ^ n)⁻¹
theorem Matrix.pow_sub' {n' : Type u_1} [] [Fintype n'] {R : Type u_2} [] (A : Matrix n' n' R) {m : } {n : } (ha : IsUnit A.det) (h : n m) :
A ^ (m - n) = A ^ m * (A ^ n)⁻¹
theorem Matrix.pow_inv_comm' {n' : Type u_1} [] [Fintype n'] {R : Type u_2} [] (A : Matrix n' n' R) (m : ) (n : ) :
A⁻¹ ^ m * A ^ n = A ^ n * A⁻¹ ^ m
@[simp]
theorem Matrix.one_zpow {n' : Type u_1} [] [Fintype n'] {R : Type u_2} [] (n : ) :
1 ^ n = 1
theorem Matrix.zero_zpow {n' : Type u_1} [] [Fintype n'] {R : Type u_2} [] (z : ) :
z 00 ^ z = 0
theorem Matrix.zero_zpow_eq {n' : Type u_1} [] [Fintype n'] {R : Type u_2} [] (n : ) :
0 ^ n = if n = 0 then 1 else 0
theorem Matrix.inv_zpow {n' : Type u_1} [] [Fintype n'] {R : Type u_2} [] (A : Matrix n' n' R) (n : ) :
A⁻¹ ^ n = (A ^ n)⁻¹
@[simp]
theorem Matrix.zpow_neg_one {n' : Type u_1} [] [Fintype n'] {R : Type u_2} [] (A : Matrix n' n' R) :
A ^ (-1) = A⁻¹
@[simp]
theorem Matrix.zpow_neg_natCast {n' : Type u_1} [] [Fintype n'] {R : Type u_2} [] (A : Matrix n' n' R) (n : ) :
A ^ (-n) = (A ^ n)⁻¹
@[deprecated Matrix.zpow_neg_natCast]
theorem Matrix.zpow_neg_coe_nat {n' : Type u_1} [] [Fintype n'] {R : Type u_2} [] (A : Matrix n' n' R) (n : ) :
A ^ (-n) = (A ^ n)⁻¹

Alias of Matrix.zpow_neg_natCast.

theorem IsUnit.det_zpow {n' : Type u_1} [] [Fintype n'] {R : Type u_2} [] {A : Matrix n' n' R} (h : IsUnit A.det) (n : ) :
IsUnit (A ^ n).det
theorem Matrix.isUnit_det_zpow_iff {n' : Type u_1} [] [Fintype n'] {R : Type u_2} [] {A : Matrix n' n' R} {z : } :
IsUnit (A ^ z).det IsUnit A.det z = 0
theorem Matrix.zpow_neg {n' : Type u_1} [] [Fintype n'] {R : Type u_2} [] {A : Matrix n' n' R} (h : IsUnit A.det) (n : ) :
A ^ (-n) = (A ^ n)⁻¹
theorem Matrix.inv_zpow' {n' : Type u_1} [] [Fintype n'] {R : Type u_2} [] {A : Matrix n' n' R} (h : IsUnit A.det) (n : ) :
A⁻¹ ^ n = A ^ (-n)
theorem Matrix.zpow_add_one {n' : Type u_1} [] [Fintype n'] {R : Type u_2} [] {A : Matrix n' n' R} (h : IsUnit A.det) (n : ) :
A ^ (n + 1) = A ^ n * A
theorem Matrix.zpow_sub_one {n' : Type u_1} [] [Fintype n'] {R : Type u_2} [] {A : Matrix n' n' R} (h : IsUnit A.det) (n : ) :
A ^ (n - 1) = A ^ n * A⁻¹
theorem Matrix.zpow_add {n' : Type u_1} [] [Fintype n'] {R : Type u_2} [] {A : Matrix n' n' R} (ha : IsUnit A.det) (m : ) (n : ) :
A ^ (m + n) = A ^ m * A ^ n
theorem Matrix.zpow_add_of_nonpos {n' : Type u_1} [] [Fintype n'] {R : Type u_2} [] {A : Matrix n' n' R} {m : } {n : } (hm : m 0) (hn : n 0) :
A ^ (m + n) = A ^ m * A ^ n
theorem Matrix.zpow_add_of_nonneg {n' : Type u_1} [] [Fintype n'] {R : Type u_2} [] {A : Matrix n' n' R} {m : } {n : } (hm : 0 m) (hn : 0 n) :
A ^ (m + n) = A ^ m * A ^ n
theorem Matrix.zpow_one_add {n' : Type u_1} [] [Fintype n'] {R : Type u_2} [] {A : Matrix n' n' R} (h : IsUnit A.det) (i : ) :
A ^ (1 + i) = A * A ^ i
theorem Matrix.SemiconjBy.zpow_right {n' : Type u_1} [] [Fintype n'] {R : Type u_2} [] {A : Matrix n' n' R} {X : Matrix n' n' R} {Y : Matrix n' n' R} (hx : IsUnit X.det) (hy : IsUnit Y.det) (h : SemiconjBy A X Y) (m : ) :
SemiconjBy A (X ^ m) (Y ^ m)
theorem Matrix.Commute.zpow_right {n' : Type u_1} [] [Fintype n'] {R : Type u_2} [] {A : Matrix n' n' R} {B : Matrix n' n' R} (h : Commute A B) (m : ) :
Commute A (B ^ m)
theorem Matrix.Commute.zpow_left {n' : Type u_1} [] [Fintype n'] {R : Type u_2} [] {A : Matrix n' n' R} {B : Matrix n' n' R} (h : Commute A B) (m : ) :
Commute (A ^ m) B
theorem Matrix.Commute.zpow_zpow {n' : Type u_1} [] [Fintype n'] {R : Type u_2} [] {A : Matrix n' n' R} {B : Matrix n' n' R} (h : Commute A B) (m : ) (n : ) :
Commute (A ^ m) (B ^ n)
theorem Matrix.Commute.zpow_self {n' : Type u_1} [] [Fintype n'] {R : Type u_2} [] (A : Matrix n' n' R) (n : ) :
Commute (A ^ n) A
theorem Matrix.Commute.self_zpow {n' : Type u_1} [] [Fintype n'] {R : Type u_2} [] (A : Matrix n' n' R) (n : ) :
Commute A (A ^ n)
theorem Matrix.Commute.zpow_zpow_self {n' : Type u_1} [] [Fintype n'] {R : Type u_2} [] (A : Matrix n' n' R) (m : ) (n : ) :
Commute (A ^ m) (A ^ n)
theorem Matrix.zpow_add_one_of_ne_neg_one {n' : Type u_1} [] [Fintype n'] {R : Type u_2} [] {A : Matrix n' n' R} (n : ) :
n -1A ^ (n + 1) = A ^ n * A
theorem Matrix.zpow_mul {n' : Type u_1} [] [Fintype n'] {R : Type u_2} [] (A : Matrix n' n' R) (h : IsUnit A.det) (m : ) (n : ) :
A ^ (m * n) = (A ^ m) ^ n
theorem Matrix.zpow_mul' {n' : Type u_1} [] [Fintype n'] {R : Type u_2} [] (A : Matrix n' n' R) (h : IsUnit A.det) (m : ) (n : ) :
A ^ (m * n) = (A ^ n) ^ m
@[simp]
theorem Matrix.coe_units_zpow {n' : Type u_1} [] [Fintype n'] {R : Type u_2} [] (u : (Matrix n' n' R)ˣ) (n : ) :
(u ^ n) = u ^ n
theorem Matrix.zpow_ne_zero_of_isUnit_det {n' : Type u_1} [] [Fintype n'] {R : Type u_2} [] [Nonempty n'] [] {A : Matrix n' n' R} (ha : IsUnit A.det) (z : ) :
A ^ z 0
theorem Matrix.zpow_sub {n' : Type u_1} [] [Fintype n'] {R : Type u_2} [] {A : Matrix n' n' R} (ha : IsUnit A.det) (z1 : ) (z2 : ) :
A ^ (z1 - z2) = A ^ z1 / A ^ z2
theorem Matrix.Commute.mul_zpow {n' : Type u_1} [] [Fintype n'] {R : Type u_2} [] {A : Matrix n' n' R} {B : Matrix n' n' R} (h : Commute A B) (i : ) :
(A * B) ^ i = A ^ i * B ^ i
theorem Matrix.zpow_neg_mul_zpow_self {n' : Type u_1} [] [Fintype n'] {R : Type u_2} [] (n : ) {A : Matrix n' n' R} (h : IsUnit A.det) :
A ^ (-n) * A ^ n = 1
theorem Matrix.one_div_pow {n' : Type u_1} [] [Fintype n'] {R : Type u_2} [] {A : Matrix n' n' R} (n : ) :
(1 / A) ^ n = 1 / A ^ n
theorem Matrix.one_div_zpow {n' : Type u_1} [] [Fintype n'] {R : Type u_2} [] {A : Matrix n' n' R} (n : ) :
(1 / A) ^ n = 1 / A ^ n
@[simp]
theorem Matrix.transpose_zpow {n' : Type u_1} [] [Fintype n'] {R : Type u_2} [] (A : Matrix n' n' R) (n : ) :
(A ^ n).transpose = A.transpose ^ n
@[simp]
theorem Matrix.conjTranspose_zpow {n' : Type u_1} [] [Fintype n'] {R : Type u_2} [] [] (A : Matrix n' n' R) (n : ) :
(A ^ n).conjTranspose = A.conjTranspose ^ n
theorem Matrix.IsSymm.zpow {n' : Type u_1} [] [Fintype n'] {R : Type u_2} [] {A : Matrix n' n' R} (h : A.IsSymm) (k : ) :
(A ^ k).IsSymm