mathlib documentation

linear_algebra.special_linear_group

The Special Linear group $SL(n, R)$ #

This file defines the elements of the Special Linear group special_linear_group n R, consisting of all square R-matrices with determinant 1 on the fintype n by n. In addition, we define the group structure on special_linear_group n R and the embedding into the general linear group general_linear_group R (n → R).

Main definitions #

Notation #

For m : ℕ, we introduce the notation SL(m,R) for the special linear group on the fintype n = fin m, in the locale matrix_groups.

Implementation notes #

The inverse operation in the special_linear_group is defined to be the adjugate matrix, so that special_linear_group n R has a group structure for all comm_ring R.

We define the elements of special_linear_group to be matrices, since we need to compute their determinant. This is in contrast with general_linear_group R M, which consists of invertible R-linear maps on M.

We provide matrix.special_linear_group.has_coe_to_fun for convenience, but do not state any lemmas about it, and use matrix.special_linear_group.coe_fn_eq_coe to eliminate it in favor of a regular coercion.

References #

Tags #

matrix group, group, matrix inverse

def matrix.special_linear_group (n : Type u) [decidable_eq n] [fintype n] (R : Type v) [comm_ring R] :
Type (max u v)

special_linear_group n R is the group of n by n R-matrices with determinant equal to 1.

Equations
theorem matrix.special_linear_group.ext_iff {n : Type u} [decidable_eq n] [fintype n] {R : Type v} [comm_ring R] (A B : matrix.special_linear_group n R) :
A = B ∀ (i j : n), A i j = B i j
@[ext]
theorem matrix.special_linear_group.ext {n : Type u} [decidable_eq n] [fintype n] {R : Type v} [comm_ring R] (A B : matrix.special_linear_group n R) :
(∀ (i j : n), A i j = B i j)A = B
@[simp]
@[simp]
theorem matrix.special_linear_group.coe_mul {n : Type u} [decidable_eq n] [fintype n] {R : Type v} [comm_ring R] (A B : matrix.special_linear_group n R) :
A * B = A B
@[simp]
theorem matrix.special_linear_group.coe_one {n : Type u} [decidable_eq n] [fintype n] {R : Type v} [comm_ring R] :
1 = 1
@[simp]
theorem matrix.special_linear_group.det_coe {n : Type u} [decidable_eq n] [fintype n] {R : Type v} [comm_ring R] (A : matrix.special_linear_group n R) :
A.det = 1
theorem matrix.special_linear_group.row_ne_zero {n : Type u} [decidable_eq n] [fintype n] {R : Type v} [comm_ring R] [nontrivial R] (g : matrix.special_linear_group n R) (i : n) :
g i 0
@[instance]
Equations
@[instance]
Equations
def matrix.special_linear_group.to_lin' {n : Type u} [decidable_eq n] [fintype n] {R : Type v} [comm_ring R] :

A version of matrix.to_lin' A that produces linear equivalences.

Equations
@[instance]

Formal operation of negation on special linear group on even cardinality n given by negating each element.

Equations
@[simp]
theorem matrix.special_linear_group.coe_neg {n : Type u} [decidable_eq n] [fintype n] {R : Type v} [comm_ring R] [fact (even (fintype.card n))] (g : matrix.special_linear_group n R) :
@[instance]

This instance is here for convenience, but is not the simp-normal form.

Equations
@[simp]
theorem matrix.special_linear_group.coe_fn_eq_coe {n : Type u} [decidable_eq n] [fintype n] {R : Type v} [comm_ring R] (s : matrix.special_linear_group n R) :