mathlib documentation

data.matrix.basic

Matrices #

@[nolint]
def matrix (m : Type u) (n : Type u') [fintype m] [fintype n] (α : Type v) :
Type (max u u' v)

matrix m n is the type of matrices whose rows are indexed by the fintype m and whose columns are indexed by the fintype n.

Equations
theorem matrix.ext_iff {m : Type u_2} {n : Type u_3} [fintype m] [fintype n] {α : Type v} {M N : matrix m n α} :
(∀ (i : m) (j : n), M i j = N i j) M = N
@[ext]
theorem matrix.ext {m : Type u_2} {n : Type u_3} [fintype m] [fintype n] {α : Type v} {M N : matrix m n α} :
(∀ (i : m) (j : n), M i j = N i j)M = N
def matrix.map {m : Type u_2} {n : Type u_3} [fintype m] [fintype n] {α : Type v} (M : matrix m n α) {β : Type w} (f : α → β) :
matrix m n β

M.map f is the matrix obtained by applying f to each entry of the matrix M.

Equations
  • M.map f = λ (i : m) (j : n), f (M i j)
@[simp]
theorem matrix.map_apply {m : Type u_2} {n : Type u_3} [fintype m] [fintype n] {α : Type v} {M : matrix m n α} {β : Type w} {f : α → β} {i : m} {j : n} :
M.map f i j = f (M i j)
@[simp]
theorem matrix.map_map {m : Type u_2} {n : Type u_3} [fintype m] [fintype n] {α : Type v} {M : matrix m n α} {β : Type u_1} {γ : Type u_4} {f : α → β} {g : β → γ} :
(M.map f).map g = M.map (g f)
def matrix.transpose {m : Type u_2} {n : Type u_3} [fintype m] [fintype n] {α : Type v} (M : matrix m n α) :
matrix n m α

The transpose of a matrix.

Equations
def matrix.col {m : Type u_2} [fintype m] {α : Type v} (w : m → α) :

matrix.col u is the column matrix whose entries are given by u.

Equations
def matrix.row {n : Type u_3} [fintype n] {α : Type v} (v : n → α) :

matrix.row u is the row matrix whose entries are given by u.

Equations
@[instance]
def matrix.inhabited {m : Type u_2} {n : Type u_3} [fintype m] [fintype n] {α : Type v} [inhabited α] :
inhabited (matrix m n α)
Equations
@[instance]
def matrix.has_add {m : Type u_2} {n : Type u_3} [fintype m] [fintype n] {α : Type v} [has_add α] :
has_add (matrix m n α)
Equations
@[instance]
def matrix.add_semigroup {m : Type u_2} {n : Type u_3} [fintype m] [fintype n] {α : Type v} [add_semigroup α] :
Equations
@[instance]
def matrix.add_comm_semigroup {m : Type u_2} {n : Type u_3} [fintype m] [fintype n] {α : Type v} [add_comm_semigroup α] :
Equations
@[instance]
def matrix.has_zero {m : Type u_2} {n : Type u_3} [fintype m] [fintype n] {α : Type v} [has_zero α] :
has_zero (matrix m n α)
Equations
@[instance]
def matrix.add_monoid {m : Type u_2} {n : Type u_3} [fintype m] [fintype n] {α : Type v} [add_monoid α] :
Equations
@[instance]
def matrix.add_comm_monoid {m : Type u_2} {n : Type u_3} [fintype m] [fintype n] {α : Type v} [add_comm_monoid α] :
Equations
@[instance]
def matrix.has_neg {m : Type u_2} {n : Type u_3} [fintype m] [fintype n] {α : Type v} [has_neg α] :
has_neg (matrix m n α)
Equations
@[instance]
def matrix.has_sub {m : Type u_2} {n : Type u_3} [fintype m] [fintype n] {α : Type v} [has_sub α] :
has_sub (matrix m n α)
Equations
@[instance]
def matrix.add_group {m : Type u_2} {n : Type u_3} [fintype m] [fintype n] {α : Type v} [add_group α] :
add_group (matrix m n α)
Equations
@[instance]
def matrix.add_comm_group {m : Type u_2} {n : Type u_3} [fintype m] [fintype n] {α : Type v} [add_comm_group α] :
Equations
@[instance]
def matrix.unique {m : Type u_2} {n : Type u_3} [fintype m] [fintype n] {α : Type v} [unique α] :
unique (matrix m n α)
Equations
@[instance]
def matrix.subsingleton {m : Type u_2} {n : Type u_3} [fintype m] [fintype n] {α : Type v} [subsingleton α] :
@[instance]
def matrix.nontrivial {m : Type u_2} {n : Type u_3} [fintype m] [fintype n] {α : Type v} [nonempty m] [nonempty n] [nontrivial α] :
@[simp]
theorem matrix.zero_apply {m : Type u_2} {n : Type u_3} [fintype m] [fintype n] {α : Type v} [has_zero α] (i : m) (j : n) :
0 i j = 0
@[simp]
theorem matrix.neg_apply {m : Type u_2} {n : Type u_3} [fintype m] [fintype n] {α : Type v} [has_neg α] (M : matrix m n α) (i : m) (j : n) :
(-M) i j = -M i j
@[simp]
theorem matrix.add_apply {m : Type u_2} {n : Type u_3} [fintype m] [fintype n] {α : Type v} [has_add α] (M N : matrix m n α) (i : m) (j : n) :
(M + N) i j = M i j + N i j
@[simp]
theorem matrix.sub_apply {m : Type u_2} {n : Type u_3} [fintype m] [fintype n] {α : Type v} [has_sub α] (M N : matrix m n α) (i : m) (j : n) :
(M - N) i j = M i j - N i j
@[simp]
theorem matrix.map_zero {m : Type u_2} {n : Type u_3} [fintype m] [fintype n] {α : Type v} [has_zero α] {β : Type w} [has_zero β] {f : α → β} (h : f 0 = 0) :
0.map f = 0
theorem matrix.map_add {m : Type u_2} {n : Type u_3} [fintype m] [fintype n] {α : Type v} [add_monoid α] {β : Type w} [add_monoid β] (f : α →+ β) (M N : matrix m n α) :
(M + N).map f = M.map f + N.map f
theorem matrix.map_sub {m : Type u_2} {n : Type u_3} [fintype m] [fintype n] {α : Type v} [add_group α] {β : Type w} [add_group β] (f : α →+ β) (M N : matrix m n α) :
(M - N).map f = M.map f - N.map f
theorem matrix.subsingleton_of_empty_left {m : Type u_2} {n : Type u_3} [fintype m] [fintype n] {α : Type v} (hm : ¬nonempty m) :
theorem matrix.subsingleton_of_empty_right {m : Type u_2} {n : Type u_3} [fintype m] [fintype n] {α : Type v} (hn : ¬nonempty n) :
def add_monoid_hom.map_matrix {m : Type u_2} {n : Type u_3} [fintype m] [fintype n] {α : Type v} [add_monoid α] {β : Type w} [add_monoid β] (f : α →+ β) :
matrix m n α →+ matrix m n β

The add_monoid_hom between spaces of matrices induced by an add_monoid_hom between their coefficients.

Equations
@[simp]
theorem add_monoid_hom.map_matrix_apply {m : Type u_2} {n : Type u_3} [fintype m] [fintype n] {α : Type v} [add_monoid α] {β : Type w} [add_monoid β] (f : α →+ β) (M : matrix m n α) :
def matrix.diagonal {n : Type u_3} [fintype n] {α : Type v} [decidable_eq n] [has_zero α] (d : n → α) :
matrix n n α

diagonal d is the square matrix such that (diagonal d) i i = d i and (diagonal d) i j = 0 if i ≠ j.

Equations
@[simp]
theorem matrix.diagonal_apply_eq {n : Type u_3} [fintype n] {α : Type v} [decidable_eq n] [has_zero α] {d : n → α} (i : n) :
matrix.diagonal d i i = d i
@[simp]
theorem matrix.diagonal_apply_ne {n : Type u_3} [fintype n] {α : Type v} [decidable_eq n] [has_zero α] {d : n → α} {i j : n} (h : i j) :
theorem matrix.diagonal_apply_ne' {n : Type u_3} [fintype n] {α : Type v} [decidable_eq n] [has_zero α] {d : n → α} {i j : n} (h : j i) :
@[simp]
theorem matrix.diagonal_zero {n : Type u_3} [fintype n] {α : Type v} [decidable_eq n] [has_zero α] :
matrix.diagonal (λ (_x : n), 0) = 0
@[simp]
theorem matrix.diagonal_transpose {n : Type u_3} [fintype n] {α : Type v} [decidable_eq n] [has_zero α] (v : n → α) :
@[simp]
theorem matrix.diagonal_add {n : Type u_3} [fintype n] {α : Type v} [decidable_eq n] [add_monoid α] (d₁ d₂ : n → α) :
matrix.diagonal d₁ + matrix.diagonal d₂ = matrix.diagonal (λ (i : n), d₁ i + d₂ i)
@[simp]
theorem matrix.diagonal_map {n : Type u_3} [fintype n] {α : Type v} [decidable_eq n] {β : Type w} [has_zero α] [has_zero β] {f : α → β} (h : f 0 = 0) {d : n → α} :
(matrix.diagonal d).map f = matrix.diagonal (λ (m : n), f (d m))
@[instance]
def matrix.has_one {n : Type u_3} [fintype n] {α : Type v} [decidable_eq n] [has_zero α] [has_one α] :
has_one (matrix n n α)
Equations
@[simp]
theorem matrix.diagonal_one {n : Type u_3} [fintype n] {α : Type v} [decidable_eq n] [has_zero α] [has_one α] :
matrix.diagonal (λ (_x : n), 1) = 1
theorem matrix.one_apply {n : Type u_3} [fintype n] {α : Type v} [decidable_eq n] [has_zero α] [has_one α] {i j : n} :
1 i j = ite (i = j) 1 0
@[simp]
theorem matrix.one_apply_eq {n : Type u_3} [fintype n] {α : Type v} [decidable_eq n] [has_zero α] [has_one α] (i : n) :
1 i i = 1
@[simp]
theorem matrix.one_apply_ne {n : Type u_3} [fintype n] {α : Type v} [decidable_eq n] [has_zero α] [has_one α] {i j : n} :
i j1 i j = 0
theorem matrix.one_apply_ne' {n : Type u_3} [fintype n] {α : Type v} [decidable_eq n] [has_zero α] [has_one α] {i j : n} :
j i1 i j = 0
@[simp]
theorem matrix.one_map {n : Type u_3} [fintype n] {α : Type v} [decidable_eq n] [has_zero α] [has_one α] {β : Type w} [has_zero β] [has_one β] {f : α → β} (h₀ : f 0 = 0) (h₁ : f 1 = 1) :
1.map f = 1
@[simp]
theorem matrix.bit0_apply {m : Type u_2} [fintype m] {α : Type v} [has_add α] (M : matrix m m α) (i j : m) :
bit0 M i j = bit0 (M i j)
theorem matrix.bit1_apply {n : Type u_3} [fintype n] {α : Type v} [decidable_eq n] [add_monoid α] [has_one α] (M : matrix n n α) (i j : n) :
bit1 M i j = ite (i = j) (bit1 (M i j)) (bit0 (M i j))
@[simp]
theorem matrix.bit1_apply_eq {n : Type u_3} [fintype n] {α : Type v} [decidable_eq n] [add_monoid α] [has_one α] (M : matrix n n α) (i : n) :
bit1 M i i = bit1 (M i i)
@[simp]
theorem matrix.bit1_apply_ne {n : Type u_3} [fintype n] {α : Type v} [decidable_eq n] [add_monoid α] [has_one α] (M : matrix n n α) {i j : n} (h : i j) :
bit1 M i j = bit0 (M i j)
def matrix.dot_product {m : Type u_2} [fintype m] {α : Type v} [has_mul α] [add_comm_monoid α] (v w : m → α) :
α

dot_product v w is the sum of the entrywise products v i * w i

Equations
theorem matrix.dot_product_assoc {m : Type u_2} {n : Type u_3} [fintype m] [fintype n] {α : Type v} [semiring α] (u : m → α) (v : m → n → α) (w : n → α) :
matrix.dot_product (λ (j : n), matrix.dot_product u (λ (i : m), v i j)) w = matrix.dot_product u (λ (i : m), matrix.dot_product (v i) w)
theorem matrix.dot_product_comm {m : Type u_2} [fintype m] {α : Type v} [comm_semiring α] (v w : m → α) :
@[simp]
theorem matrix.dot_product_punit {α : Type v} [add_comm_monoid α] [has_mul α] (v w : punit → α) :
@[simp]
theorem matrix.dot_product_zero {m : Type u_2} [fintype m] {α : Type v} [semiring α] (v : m → α) :
@[simp]
theorem matrix.dot_product_zero' {m : Type u_2} [fintype m] {α : Type v} [semiring α] (v : m → α) :
matrix.dot_product v (λ (_x : m), 0) = 0
@[simp]
theorem matrix.zero_dot_product {m : Type u_2} [fintype m] {α : Type v} [semiring α] (v : m → α) :
@[simp]
theorem matrix.zero_dot_product' {m : Type u_2} [fintype m] {α : Type v} [semiring α] (v : m → α) :
matrix.dot_product (λ (_x : m), 0) v = 0
@[simp]
theorem matrix.add_dot_product {m : Type u_2} [fintype m] {α : Type v} [semiring α] (u v w : m → α) :
@[simp]
theorem matrix.dot_product_add {m : Type u_2} [fintype m] {α : Type v} [semiring α] (u v w : m → α) :
@[simp]
theorem matrix.diagonal_dot_product {m : Type u_2} [fintype m] {α : Type v} [decidable_eq m] [semiring α] (v w : m → α) (i : m) :
@[simp]
theorem matrix.dot_product_diagonal {m : Type u_2} [fintype m] {α : Type v} [decidable_eq m] [semiring α] (v w : m → α) (i : m) :
@[simp]
theorem matrix.dot_product_diagonal' {m : Type u_2} [fintype m] {α : Type v} [decidable_eq m] [semiring α] (v w : m → α) (i : m) :
matrix.dot_product v (λ (j : m), matrix.diagonal w j i) = (v i) * w i
@[simp]
theorem matrix.neg_dot_product {m : Type u_2} [fintype m] {α : Type v} [ring α] (v w : m → α) :
@[simp]
theorem matrix.dot_product_neg {m : Type u_2} [fintype m] {α : Type v} [ring α] (v w : m → α) :
@[simp]
theorem matrix.smul_dot_product {m : Type u_2} [fintype m] {α : Type v} [semiring α] (x : α) (v w : m → α) :
@[simp]
theorem matrix.dot_product_smul {m : Type u_2} [fintype m] {α : Type v} [comm_semiring α] (x : α) (v w : m → α) :
def matrix.mul {l : Type u_1} {m : Type u_2} {n : Type u_3} [fintype l] [fintype m] [fintype n] {α : Type v} [has_mul α] [add_comm_monoid α] (M : matrix l m α) (N : matrix m n α) :
matrix l n α

M ⬝ N is the usual product of matrices M and N, i.e. we have that (M ⬝ N) i k is the dot product of the i-th row of M by the k-th column of Ǹ.

Equations
theorem matrix.mul_apply {l : Type u_1} {m : Type u_2} {n : Type u_3} [fintype l] [fintype m] [fintype n] {α : Type v} [has_mul α] [add_comm_monoid α] {M : matrix l m α} {N : matrix m n α} {i : l} {k : n} :
(M N) i k = ∑ (j : m), (M i j) * N j k
@[instance]
def matrix.has_mul {n : Type u_3} [fintype n] {α : Type v} [has_mul α] [add_comm_monoid α] :
has_mul (matrix n n α)
Equations
@[simp]
theorem matrix.mul_eq_mul {n : Type u_3} [fintype n] {α : Type v} [has_mul α] [add_comm_monoid α] (M N : matrix n n α) :
M * N = M N
theorem matrix.mul_apply' {n : Type u_3} [fintype n] {α : Type v} [has_mul α] [add_comm_monoid α] {M N : matrix n n α} {i k : n} :
(M N) i k = matrix.dot_product (λ (j : n), M i j) (λ (j : n), N j k)
theorem matrix.mul_assoc {l : Type u_1} {m : Type u_2} {n : Type u_3} {o : Type u_4} [fintype l] [fintype m] [fintype n] [fintype o] {α : Type v} [semiring α] (L : matrix l m α) (M : matrix m n α) (N : matrix n o α) :
L M N = L (M N)
@[instance]
def matrix.semigroup {n : Type u_3} [fintype n] {α : Type v} [semiring α] :
semigroup (matrix n n α)
Equations
@[simp]
theorem matrix.diagonal_neg {n : Type u_3} [fintype n] {α : Type v} [decidable_eq n] [add_group α] (d : n → α) :
-matrix.diagonal d = matrix.diagonal (λ (i : n), -d i)
@[simp]
theorem matrix.mul_zero {m : Type u_2} {n : Type u_3} {o : Type u_4} [fintype m] [fintype n] [fintype o] {α : Type v} [semiring α] (M : matrix m n α) :
M 0 = 0
@[simp]
theorem matrix.zero_mul {l : Type u_1} {m : Type u_2} {n : Type u_3} [fintype l] [fintype m] [fintype n] {α : Type v} [semiring α] (M : matrix m n α) :
0 M = 0
theorem matrix.mul_add {m : Type u_2} {n : Type u_3} {o : Type u_4} [fintype m] [fintype n] [fintype o] {α : Type v} [semiring α] (L : matrix m n α) (M N : matrix n o α) :
L (M + N) = L M + L N
theorem matrix.add_mul {l : Type u_1} {m : Type u_2} {n : Type u_3} [fintype l] [fintype m] [fintype n] {α : Type v} [semiring α] (L M : matrix l m α) (N : matrix m n α) :
(L + M) N = L N + M N
@[simp]
theorem matrix.diagonal_mul {m : Type u_2} {n : Type u_3} [fintype m] [fintype n] {α : Type v} [semiring α] [decidable_eq m] (d : m → α) (M : matrix m n α) (i : m) (j : n) :
(matrix.diagonal d M) i j = (d i) * M i j
@[simp]
theorem matrix.mul_diagonal {m : Type u_2} {n : Type u_3} [fintype m] [fintype n] {α : Type v} [semiring α] [decidable_eq n] (d : n → α) (M : matrix m n α) (i : m) (j : n) :
(M matrix.diagonal d) i j = (M i j) * d j
@[simp]
theorem matrix.one_mul {m : Type u_2} {n : Type u_3} [fintype m] [fintype n] {α : Type v} [semiring α] [decidable_eq m] (M : matrix m n α) :
1 M = M
@[simp]
theorem matrix.mul_one {m : Type u_2} {n : Type u_3} [fintype m] [fintype n] {α : Type v} [semiring α] [decidable_eq n] (M : matrix m n α) :
M 1 = M
@[instance]
def matrix.monoid {n : Type u_3} [fintype n] {α : Type v} [semiring α] [decidable_eq n] :
monoid (matrix n n α)
Equations
@[simp]
theorem matrix.diagonal_mul_diagonal {n : Type u_3} [fintype n] {α : Type v} [semiring α] [decidable_eq n] (d₁ d₂ : n → α) :
matrix.diagonal d₁ matrix.diagonal d₂ = matrix.diagonal (λ (i : n), (d₁ i) * d₂ i)
theorem matrix.diagonal_mul_diagonal' {n : Type u_3} [fintype n] {α : Type v} [semiring α] [decidable_eq n] (d₁ d₂ : n → α) :
(matrix.diagonal d₁) * matrix.diagonal d₂ = matrix.diagonal (λ (i : n), (d₁ i) * d₂ i)
@[simp]
theorem matrix.map_mul {m : Type u_2} {n : Type u_3} {o : Type u_4} [fintype m] [fintype n] [fintype o] {α : Type v} [semiring α] {L : matrix m n α} {M : matrix n o α} {β : Type w} [semiring β] {f : α →+* β} :
(L M).map f = L.map f M.map f
@[simp]
theorem matrix.ring_hom_map_one {n : Type u_3} [fintype n] {α : Type v} [semiring α] [decidable_eq n] {β : Type w} [semiring β] (f : α →+* β) :
1.map f = 1

A version of one_map where f is a ring hom.

@[simp]
theorem matrix.ring_equiv_map_one {n : Type u_3} [fintype n] {α : Type v} [semiring α] [decidable_eq n] {β : Type w} [semiring β] (f : α ≃+* β) :
1.map f = 1

A version of one_map where f is a ring_equiv.

@[simp]
theorem matrix.zero_hom_map_zero {n : Type u_3} [fintype n] {α : Type v} [semiring α] {β : Type w} [has_zero β] (f : zero_hom α β) :
0.map f = 0

A version of map_zero where f is a zero_hom.

@[simp]
theorem matrix.add_monoid_hom_map_zero {n : Type u_3} [fintype n] {α : Type v} [semiring α] {β : Type w} [add_monoid β] (f : α →+ β) :
0.map f = 0

A version of map_zero where f is a add_monoid_hom.

@[simp]
theorem matrix.add_equiv_map_zero {n : Type u_3} [fintype n] {α : Type v} [semiring α] {β : Type w} [add_monoid β] (f : α ≃+ β) :
0.map f = 0

A version of map_zero where f is a add_equiv.

@[simp]
theorem matrix.linear_map_map_zero {n : Type u_3} [fintype n] {α : Type v} [semiring α] {R : Type u_1} [semiring R] {β : Type w} [add_comm_monoid β] [semimodule R α] [semimodule R β] (f : α →ₗ[R] β) :
0.map f = 0

A version of map_zero where f is a linear_map.

@[simp]
theorem matrix.linear_equiv_map_zero {n : Type u_3} [fintype n] {α : Type v} [semiring α] {R : Type u_1} [semiring R] {β : Type w} [add_comm_monoid β] [semimodule R α] [semimodule R β] (f : α ≃ₗ[R] β) :
0.map f = 0

A version of map_zero where f is a linear_equiv.

@[simp]
theorem matrix.ring_hom_map_zero {n : Type u_3} [fintype n] {α : Type v} [semiring α] {β : Type w} [semiring β] (f : α →+* β) :
0.map f = 0

A version of map_zero where f is a ring_hom.

@[simp]
theorem matrix.ring_equiv_map_zero {n : Type u_3} [fintype n] {α : Type v} [semiring α] {β : Type w} [semiring β] (f : α ≃+* β) :
0.map f = 0

A version of map_zero where f is a ring_equiv.

theorem matrix.is_add_monoid_hom_mul_left {l : Type u_1} {m : Type u_2} {n : Type u_3} [fintype l] [fintype m] [fintype n] {α : Type v} [semiring α] (M : matrix l m α) :
is_add_monoid_hom (λ (x : matrix m n α), M x)
theorem matrix.is_add_monoid_hom_mul_right {l : Type u_1} {m : Type u_2} {n : Type u_3} [fintype l] [fintype m] [fintype n] {α : Type v} [semiring α] (M : matrix m n α) :
is_add_monoid_hom (λ (x : matrix l m α), x M)
theorem matrix.sum_mul {l : Type u_1} {m : Type u_2} {n : Type u_3} [fintype l] [fintype m] [fintype n] {α : Type v} [semiring α] {β : Type u_4} (s : finset β) (f : β → matrix l m α) (M : matrix m n α) :
(∑ (a : β) in s, f a) M = ∑ (a : β) in s, f a M
theorem matrix.mul_sum {l : Type u_1} {m : Type u_2} {n : Type u_3} [fintype l] [fintype m] [fintype n] {α : Type v} [semiring α] {β : Type u_4} (s : finset β) (f : β → matrix m n α) (M : matrix l m α) :
M ∑ (a : β) in s, f a = ∑ (a : β) in s, M f a
@[simp]
theorem matrix.row_mul_col_apply {m : Type u_2} [fintype m] {α : Type v} [semiring α] (v w : m → α) (i j : unit) :
def ring_hom.map_matrix {m : Type u_2} [fintype m] {α : Type v} [decidable_eq m] [semiring α] {β : Type w} [semiring β] (f : α →+* β) :
matrix m m α →+* matrix m m β

The ring_hom between spaces of square matrices induced by a ring_hom between their coefficients.

Equations
@[simp]
theorem ring_hom.map_matrix_apply {m : Type u_2} [fintype m] {α : Type v} [decidable_eq m] [semiring α] {β : Type w} [semiring β] (f : α →+* β) (M : matrix m m α) :
@[simp]
theorem matrix.neg_mul {m : Type u_2} {n : Type u_3} {o : Type u_4} [fintype m] [fintype n] [fintype o] {α : Type v} [ring α] (M : matrix m n α) (N : matrix n o α) :
-M N = -(M N)
@[simp]
theorem matrix.mul_neg {m : Type u_2} {n : Type u_3} {o : Type u_4} [fintype m] [fintype n] [fintype o] {α : Type v} [ring α] (M : matrix m n α) (N : matrix n o α) :
M -N = -(M N)
theorem matrix.sub_mul {m : Type u_2} {n : Type u_3} {o : Type u_4} [fintype m] [fintype n] [fintype o] {α : Type v} [ring α] (M M' : matrix m n α) (N : matrix n o α) :
(M - M') N = M N - M' N
theorem matrix.mul_sub {m : Type u_2} {n : Type u_3} {o : Type u_4} [fintype m] [fintype n] [fintype o] {α : Type v} [ring α] (M : matrix m n α) (N N' : matrix n o α) :
M (N - N') = M N - M N'
@[instance]
def matrix.has_scalar {m : Type u_2} {n : Type u_3} [fintype m] [fintype n] {α : Type v} [semiring α] :
has_scalar α (matrix m n α)
Equations
@[instance]
def matrix.semimodule {m : Type u_2} {n : Type u_3} [fintype m] [fintype n] {α : Type v} {β : Type w} [semiring α] [add_comm_monoid β] [semimodule α β] :
semimodule α (matrix m n β)
Equations
@[simp]
theorem matrix.smul_apply {m : Type u_2} {n : Type u_3} [fintype m] [fintype n] {α : Type v} [semiring α] (a : α) (A : matrix m n α) (i : m) (j : n) :
(a A) i j = a * A i j
theorem matrix.smul_eq_diagonal_mul {m : Type u_2} {n : Type u_3} [fintype m] [fintype n] {α : Type v} [semiring α] [decidable_eq m] (M : matrix m n α) (a : α) :
a M = matrix.diagonal (λ (_x : m), a) M
@[simp]
theorem matrix.smul_mul {l : Type u_1} {m : Type u_2} {n : Type u_3} [fintype l] [fintype m] [fintype n] {α : Type v} [semiring α] (M : matrix m n α) (a : α) (N : matrix n l α) :
(a M) N = a M N
@[simp]
theorem matrix.mul_mul_left {m : Type u_2} {n : Type u_3} {o : Type u_4} [fintype m] [fintype n] [fintype o] {α : Type v} [semiring α] (M : matrix m n α) (N : matrix n o α) (a : α) :
(λ (i : m) (j : n), a * M i j) N = a M N
def matrix.scalar {α : Type v} [semiring α] (n : Type u) [decidable_eq n] [fintype n] :
α →+* matrix n n α

The ring homomorphism α →+* matrix n n α sending a to the diagonal matrix with a on the diagonal.

Equations
@[simp]
theorem matrix.coe_scalar {n : Type u_3} [fintype n] {α : Type v} [semiring α] [decidable_eq n] :
(matrix.scalar n) = λ (a : α), a 1
theorem matrix.scalar_apply_eq {n : Type u_3} [fintype n] {α : Type v} [semiring α] [decidable_eq n] (a : α) (i : n) :
(matrix.scalar n) a i i = a
theorem matrix.scalar_apply_ne {n : Type u_3} [fintype n] {α : Type v} [semiring α] [decidable_eq n] (a : α) (i j : n) (h : i j) :
(matrix.scalar n) a i j = 0
theorem matrix.scalar_inj {n : Type u_3} [fintype n] {α : Type v} [semiring α] [decidable_eq n] [nonempty n] {r s : α} :
theorem matrix.smul_eq_mul_diagonal {m : Type u_2} {n : Type u_3} [fintype m] [fintype n] {α : Type v} [comm_semiring α] [decidable_eq n] (M : matrix m n α) (a : α) :
a M = M matrix.diagonal (λ (_x : n), a)
@[simp]
theorem matrix.mul_smul {l : Type u_1} {m : Type u_2} {n : Type u_3} [fintype l] [fintype m] [fintype n] {α : Type v} [comm_semiring α] (M : matrix m n α) (a : α) (N : matrix n l α) :
M (a N) = a M N
@[simp]
theorem matrix.mul_mul_right {m : Type u_2} {n : Type u_3} {o : Type u_4} [fintype m] [fintype n] [fintype o] {α : Type v} [comm_semiring α] (M : matrix m n α) (N : matrix n o α) (a : α) :
(M λ (i : n) (j : o), a * N i j) = a M N
theorem matrix.scalar.commute {n : Type u_3} [fintype n] {α : Type v} [comm_semiring α] [decidable_eq n] (r : α) (M : matrix n n α) :
def matrix.vec_mul_vec {m : Type u_2} {n : Type u_3} [fintype m] [fintype n] {α : Type v} [semiring α] (w : m → α) (v : n → α) :
matrix m n α

For two vectors w and v, vec_mul_vec w v i j is defined to be w i * v j. Put another way, vec_mul_vec w v is exactly col w ⬝ row v.

Equations
def matrix.mul_vec {m : Type u_2} {n : Type u_3} [fintype m] [fintype n] {α : Type v} [semiring α] (M : matrix m n α) (v : n → α) :
m → α

mul_vec M v is the matrix-vector product of M and v, where v is seen as a column matrix. Put another way, mul_vec M v is the vector whose entries are those of M ⬝ col v (see col_mul_vec).

Equations
def matrix.vec_mul {m : Type u_2} {n : Type u_3} [fintype m] [fintype n] {α : Type v} [semiring α] (v : m → α) (M : matrix m n α) :
n → α

vec_mul v M is the vector-matrix product of v and M, where v is seen as a row matrix. Put another way, vec_mul v M is the vector whose entries are those of row v ⬝ M (see row_vec_mul).

Equations
@[instance]
def matrix.mul_vec.is_add_monoid_hom_left {m : Type u_2} {n : Type u_3} [fintype m] [fintype n] {α : Type v} [semiring α] (v : n → α) :
is_add_monoid_hom (λ (M : matrix m n α), M.mul_vec v)
theorem matrix.mul_vec_diagonal {m : Type u_2} [fintype m] {α : Type v} [semiring α] [decidable_eq m] (v w : m → α) (x : m) :
(matrix.diagonal v).mul_vec w x = (v x) * w x
theorem matrix.vec_mul_diagonal {m : Type u_2} [fintype m] {α : Type v} [semiring α] [decidable_eq m] (v w : m → α) (x : m) :
matrix.vec_mul v (matrix.diagonal w) x = (v x) * w x
@[simp]
theorem matrix.mul_vec_one {m : Type u_2} [fintype m] {α : Type v} [semiring α] [decidable_eq m] (v : m → α) :
1.mul_vec v = v
@[simp]
theorem matrix.vec_mul_one {m : Type u_2} [fintype m] {α : Type v} [semiring α] [decidable_eq m] (v : m → α) :
@[simp]
theorem matrix.mul_vec_zero {m : Type u_2} {n : Type u_3} [fintype m] [fintype n] {α : Type v} [semiring α] (A : matrix m n α) :
A.mul_vec 0 = 0
@[simp]
theorem matrix.vec_mul_zero {m : Type u_2} {n : Type u_3} [fintype m] [fintype n] {α : Type v} [semiring α] (A : matrix m n α) :
@[simp]
theorem matrix.vec_mul_vec_mul {m : Type u_2} {n : Type u_3} {o : Type u_4} [fintype m] [fintype n] [fintype o] {α : Type v} [semiring α] (v : m → α) (M : matrix m n α) (N : matrix n o α) :
@[simp]
theorem matrix.mul_vec_mul_vec {m : Type u_2} {n : Type u_3} {o : Type u_4} [fintype m] [fintype n] [fintype o] {α : Type v} [semiring α] (v : o → α) (M : matrix m n α) (N : matrix n o α) :
M.mul_vec (N.mul_vec v) = (M N).mul_vec v
theorem matrix.vec_mul_vec_eq {m : Type u_2} {n : Type u_3} [fintype m] [fintype n] {α : Type v} [semiring α] (w : m → α) (v : n → α) :
def matrix.std_basis_matrix {m : Type u_2} {n : Type u_3} [fintype m] [fintype n] {α : Type v} [semiring α] [decidable_eq m] [decidable_eq n] (i : m) (j : n) (a : α) :
matrix m n α

std_basis_matrix i j a is the matrix with a in the i-th row, j-th column, and zeroes elsewhere.

Equations
@[simp]
theorem matrix.smul_std_basis_matrix {m : Type u_2} {n : Type u_3} [fintype m] [fintype n] {α : Type v} [semiring α] [decidable_eq m] [decidable_eq n] (i : m) (j : n) (a b : α) :
@[simp]
theorem matrix.std_basis_matrix_zero {m : Type u_2} {n : Type u_3} [fintype m] [fintype n] {α : Type v} [semiring α] [decidable_eq m] [decidable_eq n] (i : m) (j : n) :
theorem matrix.std_basis_matrix_add {m : Type u_2} {n : Type u_3} [fintype m] [fintype n] {α : Type v} [semiring α] [decidable_eq m] [decidable_eq n] (i : m) (j : n) (a b : α) :
theorem matrix.matrix_eq_sum_std_basis {m : Type u_2} {n : Type u_3} [fintype m] [fintype n] {α : Type v} [semiring α] [decidable_eq m] [decidable_eq n] (x : matrix n m α) :
x = ∑ (i : n) (j : m), matrix.std_basis_matrix i j (x i j)
theorem matrix.std_basis_eq_basis_mul_basis {m : Type u_2} {n : Type u_3} [fintype m] [fintype n] [decidable_eq m] [decidable_eq n] (i : m) (j : n) :
matrix.std_basis_matrix i j 1 = matrix.vec_mul_vec (λ (i' : m), ite (i = i') 1 0) (λ (j' : n), ite (j = j') 1 0)
theorem matrix.induction_on' {n : Type u_3} [fintype n] [decidable_eq n] {X : Type u_1} [semiring X] {M : matrix n n X → Prop} (m : matrix n n X) (h_zero : M 0) (h_add : ∀ (p q : matrix n n X), M pM qM (p + q)) (h_std_basis : ∀ (i j : n) (x : X), M (matrix.std_basis_matrix i j x)) :
M m
theorem matrix.induction_on {n : Type u_3} [fintype n] [decidable_eq n] [nonempty n] {X : Type u_1} [semiring X] {M : matrix n n X → Prop} (m : matrix n n X) (h_add : ∀ (p q : matrix n n X), M pM qM (p + q)) (h_std_basis : ∀ (i j : n) (x : X), M (matrix.std_basis_matrix i j x)) :
M m
theorem matrix.neg_vec_mul {m : Type u_2} {n : Type u_3} [fintype m] [fintype n] {α : Type v} [ring α] (v : m → α) (A : matrix m n α) :
theorem matrix.vec_mul_neg {m : Type u_2} {n : Type u_3} [fintype m] [fintype n] {α : Type v} [ring α] (v : m → α) (A : matrix m n α) :
theorem matrix.neg_mul_vec {m : Type u_2} {n : Type u_3} [fintype m] [fintype n] {α : Type v} [ring α] (v : n → α) (A : matrix m n α) :
(-A).mul_vec v = -A.mul_vec v
theorem matrix.mul_vec_neg {m : Type u_2} {n : Type u_3} [fintype m] [fintype n] {α : Type v} [ring α] (v : n → α) (A : matrix m n α) :
A.mul_vec (-v) = -A.mul_vec v
theorem matrix.smul_mul_vec_assoc {n : Type u_3} [fintype n] {α : Type v} [ring α] (A : matrix n n α) (b : n → α) (a : α) :
(a A).mul_vec b = a A.mul_vec b
@[simp]
theorem matrix.transpose_apply {m : Type u_2} {n : Type u_3} [fintype m] [fintype n] {α : Type v} (M : matrix m n α) (i : m) (j : n) :
M j i = M i j

Tell simp what the entries are in a transposed matrix.

Compare with mul_apply, diagonal_apply_eq, etc.

@[simp]
theorem matrix.transpose_transpose {m : Type u_2} {n : Type u_3} [fintype m] [fintype n] {α : Type v} (M : matrix m n α) :
@[simp]
theorem matrix.transpose_zero {m : Type u_2} {n : Type u_3} [fintype m] [fintype n] {α : Type v} [has_zero α] :
0 = 0
@[simp]
theorem matrix.transpose_one {n : Type u_3} [fintype n] {α : Type v} [decidable_eq n] [has_zero α] [has_one α] :
1 = 1
@[simp]
theorem matrix.transpose_add {m : Type u_2} {n : Type u_3} [fintype m] [fintype n] {α : Type v} [has_add α] (M N : matrix m n α) :
(M + N) = M + N
@[simp]
theorem matrix.transpose_sub {m : Type u_2} {n : Type u_3} [fintype m] [fintype n] {α : Type v} [add_group α] (M N : matrix m n α) :
(M - N) = M - N
@[simp]
theorem matrix.transpose_mul {l : Type u_1} {m : Type u_2} {n : Type u_3} [fintype l] [fintype m] [fintype n] {α : Type v} [comm_semiring α] (M : matrix m n α) (N : matrix n l α) :
(M N) = N M
@[simp]
theorem matrix.transpose_smul {m : Type u_2} {n : Type u_3} [fintype m] [fintype n] {α : Type v} [semiring α] (c : α) (M : matrix m n α) :
(c M) = c M
@[simp]
theorem matrix.transpose_neg {m : Type u_2} {n : Type u_3} [fintype m] [fintype n] {α : Type v} [has_neg α] (M : matrix m n α) :
(-M) = -M
theorem matrix.transpose_map {m : Type u_2} {n : Type u_3} [fintype m] [fintype n] {α : Type v} {β : Type w} {f : α → β} {M : matrix m n α} :
M.map f = (M.map f)
@[instance]
def matrix.star_ring {n : Type u_3} [fintype n] [decidable_eq n] {R : Type u_7} [semiring R] [star_ring R] :

When R is a *-(semi)ring, matrix n n R becomes a *-(semi)ring with the star operation given by taking the conjugate, and the star of each entry.

Equations
@[simp]
theorem matrix.star_apply {n : Type u_3} [fintype n] [decidable_eq n] {R : Type u_7} [semiring R] [star_ring R] (M : matrix n n R) (i j : n) :
star M i j = star (M j i)
theorem matrix.star_mul {n : Type u_3} [fintype n] [decidable_eq n] {R : Type u_7} [semiring R] [star_ring R] (M N : matrix n n R) :
star (M N) = star N star M
def matrix.minor {l : Type u_1} {m : Type u_2} {n : Type u_3} {o : Type u_4} [fintype l] [fintype m] [fintype n] [fintype o] {α : Type v} (A : matrix m n α) (row : l → m) (col : o → n) :
matrix l o α

M.minor row col is the matrix obtained by reindexing the rows and the lines of M, such that M.minor row col i j = M (row i) (col j). Note that the total number of row/colums doesn't have to be preserved.

Equations
  • A.minor row col = λ (i : l) (j : o), A (row i) (col j)
@[simp]
theorem matrix.minor_apply {l : Type u_1} {m : Type u_2} {n : Type u_3} {o : Type u_4} [fintype l] [fintype m] [fintype n] [fintype o] {α : Type v} (A : matrix m n α) (row : l → m) (col : o → n) (i : l) (j : o) :
A.minor row col i j = A (row i) (col j)
@[simp]
theorem matrix.minor_id_id {m : Type u_2} {n : Type u_3} [fintype m] [fintype n] {α : Type v} (A : matrix m n α) :
A.minor id id = A
@[simp]
theorem matrix.minor_minor {l : Type u_1} {m : Type u_2} {n : Type u_3} {o : Type u_4} [fintype l] [fintype m] [fintype n] [fintype o] {α : Type v} {l₂ : Type u_5} {o₂ : Type u_6} [fintype l₂] [fintype o₂] (A : matrix m n α) (row₁ : l → m) (col₁ : o → n) (row₂ : l₂ → l) (col₂ : o₂ → o) :
(A.minor row₁ col₁).minor row₂ col₂ = A.minor (row₁ row₂) (col₁ col₂)
@[simp]
theorem matrix.transpose_minor {l : Type u_1} {m : Type u_2} {n : Type u_3} {o : Type u_4} [fintype l] [fintype m] [fintype n] [fintype o] {α : Type v} (A : matrix m n α) (row : l → m) (col : o → n) :
(A.minor row col) = A.minor col row
theorem matrix.minor_add {l : Type u_1} {m : Type u_2} {n : Type u_3} {o : Type u_4} [fintype l] [fintype m] [fintype n] [fintype o] {α : Type v} [has_add α] (A B : matrix m n α) :
(A + B).minor = A.minor + B.minor
theorem matrix.minor_neg {l : Type u_1} {m : Type u_2} {n : Type u_3} {o : Type u_4} [fintype l] [fintype m] [fintype n] [fintype o] {α : Type v} [has_neg α] (A : matrix m n α) :
theorem matrix.minor_sub {l : Type u_1} {m : Type u_2} {n : Type u_3} {o : Type u_4} [fintype l] [fintype m] [fintype n] [fintype o] {α : Type v} [has_sub α] (A B : matrix m n α) :
(A - B).minor = A.minor - B.minor
@[simp]
theorem matrix.minor_zero {l : Type u_1} {m : Type u_2} {n : Type u_3} {o : Type u_4} [fintype l] [fintype m] [fintype n] [fintype o] {α : Type v} [has_zero α] :
0.minor = 0
theorem matrix.minor_smul {l : Type u_1} {m : Type u_2} {n : Type u_3} {o : Type u_4} [fintype l] [fintype m] [fintype n] [fintype o] {α : Type v} {R : Type u_5} [semiring R] [add_comm_monoid α] [semimodule R α] (r : R) (A : matrix m n α) :
(r A).minor = r A.minor
theorem matrix.minor_diagonal {l : Type u_1} {m : Type u_2} [fintype l] [fintype m] {α : Type v} [has_zero α] [decidable_eq m] [decidable_eq l] (d : m → α) (e : l → m) (he : function.injective e) :

If the minor doesn't repeat elements, then when applied to a diagonal matrix the result is diagonal.

theorem matrix.minor_one {l : Type u_1} {m : Type u_2} [fintype l] [fintype m] {α : Type v} [has_zero α] [has_one α] [decidable_eq m] [decidable_eq l] (e : l → m) (he : function.injective e) :
1.minor e e = 1
theorem matrix.minor_mul {l : Type u_1} {m : Type u_2} {n : Type u_3} {o : Type u_4} [fintype l] [fintype m] [fintype n] [fintype o] {α : Type v} [semiring α] {p : Type u_5} {q : Type u_6} [fintype p] [fintype q] (M : matrix m n α) (N : matrix n p α) (e₁ : l → m) (e₂ : o → n) (e₃ : q → p) (he₂ : function.bijective e₂) :
(M N).minor e₁ e₃ = M.minor e₁ e₂ N.minor e₂ e₃

simp lemmas for matrix.minors interaction with matrix.diagonal, 1, and matrix.mul for when the mappings are bundled.

@[simp]
theorem matrix.minor_diagonal_embedding {l : Type u_1} {m : Type u_2} [fintype l] [fintype m] {α : Type v} [has_zero α] [decidable_eq m] [decidable_eq l] (d : m → α) (e : l m) :
@[simp]
theorem matrix.minor_diagonal_equiv {l : Type u_1} {m : Type u_2} [fintype l] [fintype m] {α : Type v} [has_zero α] [decidable_eq m] [decidable_eq l] (d : m → α) (e : l m) :
@[simp]
theorem matrix.minor_one_embedding {l : Type u_1} {m : Type u_2} [fintype l] [fintype m] {α : Type v} [has_zero α] [has_one α] [decidable_eq m] [decidable_eq l] (e : l m) :
1.minor e e = 1
@[simp]
theorem matrix.minor_one_equiv {l : Type u_1} {m : Type u_2} [fintype l] [fintype m] {α : Type v} [has_zero α] [has_one α] [decidable_eq m] [decidable_eq l] (e : l m) :
1.minor e e = 1
theorem matrix.minor_mul_equiv {l : Type u_1} {m : Type u_2} {n : Type u_3} {o : Type u_4} [fintype l] [fintype m] [fintype n] [fintype o] {α : Type v} [semiring α] {p : Type u_5} {q : Type u_6} [fintype p] [fintype q] (M : matrix m n α) (N : matrix n p α) (e₁ : l → m) (e₂ : o n) (e₃ : q → p) :
(M N).minor e₁ e₃ = M.minor e₁ e₂ N.minor e₂ e₃
def matrix.reindex {l : Type u_1} {m : Type u_2} {n : Type u_3} {o : Type u_4} [fintype l] [fintype m] [fintype n] [fintype o] {α : Type v} (eₘ : m l) (eₙ : n o) :
matrix m n α matrix l o α

The natural map that reindexes a matrix's rows and columns with equivalent types is an equivalence.

Equations
@[simp]
theorem matrix.reindex_apply {l : Type u_1} {m : Type u_2} {n : Type u_3} {o : Type u_4} [fintype l] [fintype m] [fintype n] [fintype o] {α : Type v} (eₘ : m l) (eₙ : n o) (M : matrix m n α) :
(matrix.reindex eₘ eₙ) M = M.minor (eₘ.symm) (eₙ.symm)
@[simp]
theorem matrix.reindex_refl_refl {m : Type u_2} {n : Type u_3} [fintype m] [fintype n] {α : Type v} (A : matrix m n α) :
@[simp]
theorem matrix.reindex_symm {l : Type u_1} {m : Type u_2} {n : Type u_3} {o : Type u_4} [fintype l] [fintype m] [fintype n] [fintype o] {α : Type v} (eₘ : m l) (eₙ : n o) :
@[simp]
theorem matrix.reindex_trans {l : Type u_1} {m : Type u_2} {n : Type u_3} {o : Type u_4} [fintype l] [fintype m] [fintype n] [fintype o] {α : Type v} {l₂ : Type u_5} {o₂ : Type u_6} [fintype l₂] [fintype o₂] (eₘ : m l) (eₙ : n o) (eₘ₂ : l l₂) (eₙ₂ : o o₂) :
(matrix.reindex eₘ eₙ).trans (matrix.reindex eₘ₂ eₙ₂) = matrix.reindex (eₘ.trans eₘ₂) (eₙ.trans eₙ₂)
theorem matrix.transpose_reindex {l : Type u_1} {m : Type u_2} {n : Type u_3} {o : Type u_4} [fintype l] [fintype m] [fintype n] [fintype o] {α : Type v} (eₘ : m l) (eₙ : n o) (M : matrix m n α) :
((matrix.reindex eₘ eₙ) M) = (matrix.reindex eₙ eₘ) M
def matrix.sub_left {α : Type v} {m l r : } (A : matrix (fin m) (fin (l + r)) α) :
matrix (fin m) (fin l) α

The left n × l part of a n × (l+r) matrix.

Equations
def matrix.sub_right {α : Type v} {m l r : } (A : matrix (fin m) (fin (l + r)) α) :
matrix (fin m) (fin r) α

The right n × r part of a n × (l+r) matrix.

Equations
def matrix.sub_up {α : Type v} {d u n : } (A : matrix (fin (u + d)) (fin n) α) :
matrix (fin u) (fin n) α

The top u × n part of a (u+d) × n matrix.

Equations
def matrix.sub_down {α : Type v} {d u n : } (A : matrix (fin (u + d)) (fin n) α) :
matrix (fin d) (fin n) α

The bottom d × n part of a (u+d) × n matrix.

Equations
def matrix.sub_up_right {α : Type v} {d u l r : } (A : matrix (fin (u + d)) (fin (l + r)) α) :
matrix (fin u) (fin r) α

The top-right u × r part of a (u+d) × (l+r) matrix.

Equations
def matrix.sub_down_right {α : Type v} {d u l r : } (A : matrix (fin (u + d)) (fin (l + r)) α) :
matrix (fin d) (fin r) α

The bottom-right d × r part of a (u+d) × (l+r) matrix.

Equations
def matrix.sub_up_left {α : Type v} {d u l r : } (A : matrix (fin (u + d)) (fin (l + r)) α) :
matrix (fin u) (fin l) α

The top-left u × l part of a (u+d) × (l+r) matrix.

Equations
def matrix.sub_down_left {α : Type v} {d u l r : } (A : matrix (fin (u + d)) (fin (l + r)) α) :
matrix (fin d) (fin l) α

The bottom-left d × l part of a (u+d) × (l+r) matrix.

Equations

row_col section #

Simplification lemmas for matrix.row and matrix.col.

@[simp]
theorem matrix.col_add {m : Type u_2} [fintype m] {α : Type v} [semiring α] (v w : m → α) :
@[simp]
theorem matrix.col_smul {m : Type u_2} [fintype m] {α : Type v} [semiring α] (x : α) (v : m → α) :
@[simp]
theorem matrix.row_add {m : Type u_2} [fintype m] {α : Type v} [semiring α] (v w : m → α) :
@[simp]
theorem matrix.row_smul {m : Type u_2} [fintype m] {α : Type v} [semiring α] (x : α) (v : m → α) :
@[simp]
theorem matrix.col_apply {m : Type u_2} [fintype m] {α : Type v} (v : m → α) (i : m) (j : unit) :
matrix.col v i j = v i
@[simp]
theorem matrix.row_apply {m : Type u_2} [fintype m] {α : Type v} (v : m → α) (i : unit) (j : m) :
matrix.row v i j = v j
@[simp]
theorem matrix.transpose_col {m : Type u_2} [fintype m] {α : Type v} (v : m → α) :
@[simp]
theorem matrix.transpose_row {m : Type u_2} [fintype m] {α : Type v} (v : m → α) :
theorem matrix.row_vec_mul {m : Type u_2} {n : Type u_3} [fintype m] [fintype n] {α : Type v} [semiring α] (M : matrix m n α) (v : m → α) :
theorem matrix.col_vec_mul {m : Type u_2} {n : Type u_3} [fintype m] [fintype n] {α : Type v} [semiring α] (M : matrix m n α) (v : m → α) :
theorem matrix.col_mul_vec {m : Type u_2} {n : Type u_3} [fintype m] [fintype n] {α : Type v} [semiring α] (M : matrix m n α) (v : n → α) :
theorem matrix.row_mul_vec {m : Type u_2} {n : Type u_3} [fintype m] [fintype n] {α : Type v} [semiring α] (M : matrix m n α) (v : n → α) :
def matrix.update_row {m : Type u_2} {n : Type u_3} [fintype m] [fintype n] {α : Type v} [decidable_eq n] (M : matrix n m α) (i : n) (b : m → α) :
matrix n m α

Update, i.e. replace the ith row of matrix A with the values in b.

Equations
def matrix.update_column {m : Type u_2} {n : Type u_3} [fintype m] [fintype n] {α : Type v} [decidable_eq m] (M : matrix n m α) (j : m) (b : n → α) :
matrix n m α

Update, i.e. replace the jth column of matrix A with the values in b.

Equations
@[simp]
theorem matrix.update_row_self {m : Type u_2} {n : Type u_3} [fintype m] [fintype n] {α : Type v} {M : matrix n m α} {i : n} {b : m → α} [decidable_eq n] :
M.update_row i b i = b
@[simp]
theorem matrix.update_column_self {m : Type u_2} {n : Type u_3} [fintype m] [fintype n] {α : Type v} {M : matrix n m α} {i : n} {j : m} {c : n → α} [decidable_eq m] :
M.update_column j c i j = c i
@[simp]
theorem matrix.update_row_ne {m : Type u_2} {n : Type u_3} [fintype m] [fintype n] {α : Type v} {M : matrix n m α} {i : n} {b : m → α} [decidable_eq n] {i' : n} (i_ne : i' i) :
M.update_row i b i' = M i'
@[simp]
theorem matrix.update_column_ne {m : Type u_2} {n : Type u_3} [fintype m] [fintype n] {α : Type v} {M : matrix n m α} {i : n} {j : m} {c : n → α} [decidable_eq m] {j' : m} (j_ne : j' j) :
M.update_column j c i j' = M i j'
theorem matrix.update_row_apply {m : Type u_2} {n : Type u_3} [fintype m] [fintype n] {α : Type v} {M : matrix n m α} {i : n} {j : m} {b : m → α} [decidable_eq n] {i' : n} :
M.update_row i b i' j = ite (i' = i) (b j) (M i' j)
theorem matrix.update_column_apply {m : Type u_2} {n : Type u_3} [fintype m] [fintype n] {α : Type v} {M : matrix n m α} {i : n} {j : m} {c : n → α} [decidable_eq m] {j' : m} :
M.update_column j c i j' = ite (j' = j) (c i) (M i j')
theorem matrix.update_row_transpose {m : Type u_2} {n : Type u_3} [fintype m] [fintype n] {α : Type v} {M : matrix n m α} {j : m} {c : n → α} [decidable_eq m] :
theorem matrix.update_column_transpose {m : Type u_2} {n : Type u_3} [fintype m] [fintype n] {α : Type v} {M : matrix n m α} {i : n} {b : m → α} [decidable_eq n] :
def matrix.from_blocks {l : Type u_1} {m : Type u_2} {n : Type u_3} {o : Type u_4} [fintype l] [fintype m] [fintype n] [fintype o] {α : Type v} (A : matrix n l α) (B : matrix n m α) (C : matrix o l α) (D : matrix o m α) :
matrix (n o) (l m) α

We can form a single large matrix by flattening smaller 'block' matrices of compatible dimensions.

Equations
@[simp]
theorem matrix.from_blocks_apply₁₁ {l : Type u_1} {m : Type u_2} {n : Type u_3} {o : Type u_4} [fintype l] [fintype m] [fintype n] [fintype o] {α : Type v} (A : matrix n l α) (B : matrix n m α) (C : matrix o l α) (D : matrix o m α) (i : n) (j : l) :
A.from_blocks B C D (sum.inl i) (sum.inl j) = A i j
@[simp]
theorem matrix.from_blocks_apply₁₂ {l : Type u_1} {m : Type u_2} {n : Type u_3} {o : Type u_4} [fintype l] [fintype m] [fintype n] [fintype o] {α : Type v} (A : matrix n l α) (B : matrix n m α) (C : matrix o l α) (D : matrix o m α) (i : n) (j : m) :
A.from_blocks B C D (sum.inl i) (sum.inr j) = B i j
@[simp]
theorem matrix.from_blocks_apply₂₁ {l : Type u_1} {m : Type u_2} {n : Type u_3} {o : Type u_4} [fintype l] [fintype m] [fintype n] [fintype o] {α : Type v} (A : matrix n l α) (B : matrix n m α) (C : matrix o l α) (D : matrix o m α) (i : o) (j : l) :
A.from_blocks B C D (sum.inr i) (sum.inl j) = C i j
@[simp]
theorem matrix.from_blocks_apply₂₂ {l : Type u_1} {m : Type u_2} {n : Type u_3} {o : Type u_4} [fintype l] [fintype m] [fintype n] [fintype o] {α : Type v} (A : matrix n l α) (B : matrix n m α) (C : matrix o l α) (D : matrix o m α) (i : o) (j : m) :
A.from_blocks B C D (sum.inr i) (sum.inr j) = D i j
def matrix.to_blocks₁₁ {l : Type u_1} {m : Type u_2} {n : Type u_3} {o : Type u_4} [fintype l] [fintype m] [fintype n] [fintype o] {α : Type v} (M : matrix (n o) (l m) α) :
matrix n l α

Given a matrix whose row and column indexes are sum types, we can extract the corresponding "top left" submatrix.

Equations
def matrix.to_blocks₁₂ {l : Type u_1} {m : Type u_2} {n : Type u_3} {o : Type u_4} [fintype l] [fintype m] [fintype n] [fintype o] {α : Type v} (M : matrix (n o) (l m) α) :
matrix n m α

Given a matrix whose row and column indexes are sum types, we can extract the corresponding "top right" submatrix.

Equations
def matrix.to_blocks₂₁ {l : Type u_1} {m : Type u_2} {n : Type u_3} {o : Type u_4} [fintype l] [fintype m] [fintype n] [fintype o] {α : Type v} (M : matrix (n o) (l m) α) :
matrix o l α

Given a matrix whose row and column indexes are sum types, we can extract the corresponding "bottom left" submatrix.

Equations
def matrix.to_blocks₂₂ {l : Type u_1} {m : Type u_2} {n : Type u_3} {o : Type u_4} [fintype l] [fintype m] [fintype n] [fintype o] {α : Type v} (M : matrix (n o) (l m) α) :
matrix o m α

Given a matrix whose row and column indexes are sum types, we can extract the corresponding "bottom right" submatrix.

Equations
theorem matrix.from_blocks_to_blocks {l : Type u_1} {m : Type u_2} {n : Type u_3} {o : Type u_4} [fintype l] [fintype m] [fintype n] [fintype o] {α : Type v} (M : matrix (n o) (l m) α) :
@[simp]
theorem matrix.to_blocks_from_blocks₁₁ {l : Type u_1} {m : Type u_2} {n : Type u_3} {o : Type u_4} [fintype l] [fintype m] [fintype n] [fintype o] {α : Type v} (A : matrix n l α) (B : matrix n m α) (C : matrix o l α) (D : matrix o m α) :
@[simp]
theorem matrix.to_blocks_from_blocks₁₂ {l : Type u_1} {m : Type u_2} {n : Type u_3} {o : Type u_4} [fintype l] [fintype m] [fintype n] [fintype o] {α : Type v} (A : matrix n l α) (B : matrix n m α) (C : matrix o l α) (D : matrix o m α) :
@[simp]
theorem matrix.to_blocks_from_blocks₂₁ {l : Type u_1} {m : Type u_2} {n : Type u_3} {o : Type u_4} [fintype l] [fintype m] [fintype n] [fintype o] {α : Type v} (A : matrix n l α) (B : matrix n m α) (C : matrix o l α) (D : matrix o m α) :
@[simp]
theorem matrix.to_blocks_from_blocks₂₂ {l : Type u_1} {m : Type u_2} {n : Type u_3} {o : Type u_4} [fintype l] [fintype m] [fintype n] [fintype o] {α : Type v} (A : matrix n l α) (B : matrix n m α) (C : matrix o l α) (D : matrix o m α) :
theorem matrix.from_blocks_transpose {l : Type u_1} {m : Type u_2} {n : Type u_3} {o : Type u_4} [fintype l] [fintype m] [fintype n] [fintype o] {α : Type v} (A : matrix n l α) (B : matrix n m α) (C : matrix o l α) (D : matrix o m α) :
def matrix.to_block {m : Type u_2} {n : Type u_3} [fintype m] [fintype n] {α : Type v} (M : matrix m n α) (p : m → Prop) [decidable_pred p] (q : n → Prop) [decidable_pred q] :
matrix {a // p a} {a // q a} α

Let p pick out certain rows and q pick out certain columns of a matrix M. Then to_block M p q is the corresponding block matrix.

Equations
@[simp]
theorem matrix.to_block_apply {m : Type u_2} {n : Type u_3} [fintype m] [fintype n] {α : Type v} (M : matrix m n α) (p : m → Prop) [decidable_pred p] (q : n → Prop) [decidable_pred q] (i : {a // p a}) (j : {a // q a}) :
M.to_block p q i j = M i j
def matrix.to_square_block {m : Type u_2} [fintype m] {α : Type v} (M : matrix m m α) {n : } (b : m → fin n) (k : fin n) :
matrix {a // b a = k} {a // b a = k} α

Let b map rows and columns of a square matrix M to blocks. Then to_square_block M b k is the block k matrix.

Equations
@[simp]
theorem matrix.to_square_block_def {m : Type u_2} [fintype m] {α : Type v} (M : matrix m m α) {n : } (b : m → fin n) (k : fin n) :
M.to_square_block b k = λ (i j : {a // b a = k}), M i j
def matrix.to_square_block' {m : Type u_2} [fintype m] {α : Type v} (M : matrix m m α) (b : m → ) (k : ) :
matrix {a // b a = k} {a // b a = k} α

Alternate version with b : m → nat. Let b map rows and columns of a square matrix M to blocks. Then to_square_block' M b k is the block k matrix.

Equations
@[simp]
theorem matrix.to_square_block_def' {m : Type u_2} [fintype m] {α : Type v} (M : matrix m m α) (b : m → ) (k : ) :
M.to_square_block' b k = λ (i j : {a // b a = k}), M i j
def matrix.to_square_block_prop {m : Type u_2} [fintype m] {α : Type v} (M : matrix m m α) (p : m → Prop) [decidable_pred p] :
matrix {a // p a} {a // p a} α

Let p pick out certain rows and columns of a square matrix M. Then to_square_block_prop M p is the corresponding block matrix.

Equations
@[simp]
theorem matrix.to_square_block_prop_def {m : Type u_2} [fintype m] {α : Type v} (M : matrix m m α) (p : m → Prop) [decidable_pred p] :
M.to_square_block_prop p = λ (i j : {a // p a}), M i j
theorem matrix.from_blocks_smul {l : Type u_1} {m : Type u_2} {n : Type u_3} {o : Type u_4} [fintype l] [fintype m] [fintype n] [fintype o] {α : Type v} [semiring α] (x : α) (A : matrix n l α) (B : matrix n m α) (C : matrix o l α) (D : matrix o m α) :
x A.from_blocks B C D = (x A).from_blocks (x B) (x C) (x D)
theorem matrix.from_blocks_add {l : Type u_1} {m : Type u_2} {n : Type u_3} {o : Type u_4} [fintype l] [fintype m] [fintype n] [fintype o] {α : Type v} [semiring α] (A : matrix n l α) (B : matrix n m α) (C : matrix o l α) (D : matrix o m α) (A' : matrix n l α) (B' : matrix n m α) (C' : matrix o l α) (D' : matrix o m α) :
A.from_blocks B C D + A'.from_blocks B' C' D' = (A + A').from_blocks (B + B') (C + C') (D + D')
theorem matrix.from_blocks_multiply {l : Type u_1} {m : Type u_2} {n : Type u_3} {o : Type u_4} [fintype l] [fintype m] [fintype n] [fintype o] {α : Type v} [semiring α] {p : Type u_5} {q : Type u_6} [fintype p] [fintype q] (A : matrix n l α) (B : matrix n m α) (C : matrix o l α) (D : matrix o m α) (A' : matrix l p α) (B' : matrix l q α) (C' : matrix m p α) (D' : matrix m q α) :
A.from_blocks B C D A'.from_blocks B' C' D' = (A A' + B C').from_blocks (A B' + B D') (C A' + D C') (C B' + D D')
@[simp]
theorem matrix.from_blocks_diagonal {l : Type u_1} {m : Type u_2} [fintype l] [fintype m] {α : Type v} [semiring α] [decidable_eq l] [decidable_eq m] (d₁ : l → α) (d₂ : m → α) :
@[simp]
theorem matrix.from_blocks_one {l : Type u_1} {m : Type u_2} [fintype l] [fintype m] {α : Type v} [semiring α] [decidable_eq l] [decidable_eq m] :
1.from_blocks 0 0 1 = 1
def matrix.block_diagonal {m : Type u_2} {n : Type u_3} {o : Type u_4} [fintype m] [fintype n] [fintype o] {α : Type v} (M : o → matrix m n α) [decidable_eq o] [has_zero α] :
matrix (m × o) (n × o) α

matrix.block_diagonal M turns a homogenously-indexed collection of matrices M : o → matrix m n α' into a m × o-by-n × o block matrix which has the entries of M along the diagonal and zero elsewhere.

See also matrix.block_diagonal' if the matrices may not have the same size everywhere.

Equations
theorem matrix.block_diagonal_apply {m : Type u_2} {n : Type u_3} {o : Type u_4} [fintype m] [fintype n] [fintype o] {α : Type v} (M : o → matrix m n α) [decidable_eq o] [has_zero α] (ik : m × o) (jk : n × o) :
matrix.block_diagonal M ik jk = ite (ik.snd = jk.snd) (M ik.snd ik.fst jk.fst) 0
@[simp]
theorem matrix.block_diagonal_apply_eq {m : Type u_2} {n : Type u_3} {o : Type u_4} [fintype m] [fintype n] [fintype o] {α : Type v} (M : o → matrix m n α) [decidable_eq o] [has_zero α] (i : m) (j : n) (k : o) :
matrix.block_diagonal M (i, k) (j, k) = M k i j
theorem matrix.block_diagonal_apply_ne {m : Type u_2} {n : Type u_3} {o : Type u_4} [fintype m] [fintype n] [fintype o] {α : Type v} (M : o → matrix m n α) [decidable_eq o] [has_zero α] (i : m) (j : n) {k k' : o} (h : k k') :
matrix.block_diagonal M (i, k) (j, k') = 0
@[simp]
theorem matrix.block_diagonal_transpose {m : Type u_2} {n : Type u_3} {o : Type u_4} [fintype m] [fintype n] [fintype o] {α : Type v} (M : o → matrix m n α) [decidable_eq o] [has_zero α] :
@[simp]
theorem matrix.block_diagonal_zero {m : Type u_2} {n : Type u_3} {o : Type u_4} [fintype m] [fintype n] [fintype o] {α : Type v} [decidable_eq o] [has_zero α] :
@[simp]
theorem matrix.block_diagonal_diagonal {m : Type u_2} {o : Type u_4} [fintype m] [fintype o] {α : Type v} [decidable_eq o] [has_zero α] [decidable_eq m] (d : o → m → α) :
matrix.block_diagonal (λ (k : o), matrix.diagonal (d k)) = matrix.diagonal (λ (ik : m × o), d ik.snd ik.fst)
@[simp]
theorem matrix.block_diagonal_one {m : Type u_2} {o : Type u_4} [fintype m] [fintype o] {α : Type v} [decidable_eq o] [has_zero α] [decidable_eq m] [has_one α] :
@[simp]
theorem matrix.block_diagonal_add {m : Type u_2} {n : Type u_3} {o : Type u_4} [fintype m] [fintype n] [fintype o] {α : Type v} (M N : o → matrix m n α) [decidable_eq o] [add_monoid α] :
@[simp]
theorem matrix.block_diagonal_neg {m : Type u_2} {n : Type u_3} {o : Type u_4} [fintype m] [fintype n] [fintype o] {α : Type v} (M : o → matrix m n α) [decidable_eq o] [add_group α] :
@[simp]
theorem matrix.block_diagonal_sub {m : Type u_2} {n : Type u_3} {o : Type u_4} [fintype m] [fintype n] [fintype o] {α : Type v} (M N : o → matrix m n α) [decidable_eq o] [add_group α] :
@[simp]
theorem matrix.block_diagonal_mul {m : Type u_2} {n : Type u_3} {o : Type u_4} [fintype m] [fintype n] [fintype o] {α : Type v} (M : o → matrix m n α) [decidable_eq o] {p : Type u_1} [fintype p] [semiring α] (N : o → matrix n p α) :
@[simp]
theorem matrix.block_diagonal_smul {m : Type u_2} {n : Type u_3} {o : Type u_4} [fintype m] [fintype n] [fintype o] {α : Type v} (M : o → matrix m n α) [decidable_eq o] {R : Type u_1} [semiring R] [add_comm_monoid α] [semimodule R α] (x : R) :
def matrix.block_diagonal' {o : Type u_4} [fintype o] {m' : o → Type u_5} [Π (i : o), fintype (m' i)] {n' : o → Type u_6} [Π (i : o), fintype (n' i)] {α : Type v} (M : Π (i : o), matrix (m' i) (n' i) α) [decidable_eq o] [has_zero α] :
matrix (Σ (i : o), m' i) (Σ (i : o), n' i) α

matrix.block_diagonal' M turns M : Π i, matrix (m i) (n i) α into a Σ i, m i-by-Σ i, n i block matrix which has the entries of M along the diagonal and zero elsewhere.

This is the dependently-typed version of matrix.block_diagonal.

Equations
theorem matrix.block_diagonal'_eq_block_diagonal {m : Type u_2} {n : Type u_3} {o : Type u_4} [fintype m] [fintype n] [fintype o] {α : Type v} [decidable_eq o] [has_zero α] (M : o → matrix m n α) {k k' : o} (i : m) (j : n) :
matrix.block_diagonal M (i, k) (j, k') = matrix.block_diagonal' M k, i⟩ k', j⟩
theorem matrix.block_diagonal'_minor_eq_block_diagonal {m : Type u_2} {n : Type u_3} {o : Type u_4} [fintype m] [fintype n] [fintype o] {α : Type v} [decidable_eq o] [has_zero α] (M : o → matrix m n α) :
theorem matrix.block_diagonal'_apply {o : Type u_4} [fintype o] {m' : o → Type u_5} [Π (i : o), fintype (m' i)] {n' : o → Type u_6} [Π (i : o), fintype (n' i)] {α : Type v} (M : Π (i : o), matrix (m' i) (n' i) α) [decidable_eq o] [has_zero α] (ik : Σ (i : o), m' i) (jk : Σ (i : o), n' i) :
matrix.block_diagonal' M ik jk = dite (ik.fst = jk.fst) (λ (h : ik.fst = jk.fst), M ik.fst ik.snd (cast _ jk.snd)) (λ (h : ¬ik.fst = jk.fst), 0)
@[simp]
theorem matrix.block_diagonal'_apply_eq {o : Type u_4} [fintype o] {m' : o → Type u_5} [Π (i : o), fintype (m' i)] {n' : o → Type u_6} [Π (i : o), fintype (n' i)] {α : Type v} (M : Π (i : o), matrix (m' i) (n' i) α) [decidable_eq o] [has_zero α] (k : o) (i : m' k) (j : n' k) :
matrix.block_diagonal' M k, i⟩ k, j⟩ = M k i j
theorem matrix.block_diagonal'_apply_ne {o : Type u_4} [fintype o] {m' : o → Type u_5} [Π (i : o), fintype (m' i)] {n' : o → Type u_6} [Π (i : o), fintype (n' i)] {α : Type v} (M : Π (i : o), matrix (m' i) (n' i) α) [decidable_eq o] [has_zero α] {k k' : o} (i : m' k) (j : n' k') (h : k k') :
matrix.block_diagonal' M k, i⟩ k', j⟩ = 0
@[simp]
theorem matrix.block_diagonal'_transpose {o : Type u_4} [fintype o] {m' : o → Type u_5} [Π (i : o), fintype (m' i)] {n' : o → Type u_6} [Π (i : o), fintype (n' i)] {α : Type v} (M : Π (i : o), matrix (m' i) (n' i) α) [decidable_eq o] [has_zero α] :
@[simp]
theorem matrix.block_diagonal'_zero {o : Type u_4} [fintype o] {m' : o → Type u_5} [Π (i : o), fintype (m' i)] {n' : o → Type u_6} [Π (i : o), fintype (n' i)] {α : Type v} [decidable_eq o] [has_zero α] :
@[simp]
theorem matrix.block_diagonal'_diagonal {o : Type u_4} [fintype o] {m' : o → Type u_5} [Π (i : o), fintype (m' i)] {α : Type v} [decidable_eq o] [has_zero α] [Π (i : o), decidable_eq (m' i)] (d : Π (i : o), m' i → α) :
matrix.block_diagonal' (λ (k : o), matrix.diagonal (d k)) = matrix.diagonal (λ (ik : Σ (i : o), m' i), d ik.fst ik.snd)
@[simp]
theorem matrix.block_diagonal'_one {o : Type u_4} [fintype o] {m' : o → Type u_5} [Π (i : o), fintype (m' i)] {α : Type v} [decidable_eq o] [has_zero α] [Π (i : o), decidable_eq (m' i)] [has_one α] :
@[simp]
theorem matrix.block_diagonal'_add {o : Type u_4} [fintype o] {m' : o → Type u_5} [Π (i : o), fintype (m' i)] {n' : o → Type u_6} [Π (i : o), fintype (n' i)] {α : Type v} (M N : Π (i : o), matrix (m' i) (n' i) α) [decidable_eq o] [add_monoid α] :
@[simp]
theorem matrix.block_diagonal'_neg {o : Type u_4} [fintype o] {m' : o → Type u_5} [Π (i : o), fintype (m' i)] {n' : o → Type u_6} [Π (i : o), fintype (n' i)] {α : Type v} (M : Π (i : o), matrix (m' i) (n' i) α) [decidable_eq o] [add_group α] :
@[simp]
theorem matrix.block_diagonal'_sub {o : Type u_4} [fintype o] {m' : o → Type u_5} [Π (i : o), fintype (m' i)] {n' : o → Type u_6} [Π (i : o), fintype (n' i)] {α : Type v} (M N : Π (i : o), matrix (m' i) (n' i) α) [decidable_eq o] [add_group α] :
@[simp]
theorem matrix.block_diagonal'_mul {o : Type u_4} [fintype o] {m' : o → Type u_5} [Π (i : o), fintype (m' i)] {n' : o → Type u_6} [Π (i : o), fintype (n' i)] {α : Type v} (M : Π (i : o), matrix (m' i) (n' i) α) [decidable_eq o] {p : o → Type u_1} [Π (i : o), fintype (p i)] [semiring α] (N : Π (i : o), matrix (n' i) (p i) α) :
@[simp]
theorem matrix.block_diagonal'_smul {o : Type u_4} [fintype o] {m' : o → Type u_5} [Π (i : o), fintype (m' i)] {n' : o → Type u_6} [Π (i : o), fintype (n' i)] {α : Type v} (M : Π (i : o), matrix (m' i) (n' i) α) [decidable_eq o] {R : Type u_1} [semiring R] [add_comm_monoid α] [semimodule R α] (x : R) :
theorem ring_hom.map_matrix_mul {m : Type u_2} {n : Type u_3} {o : Type u_4} [fintype m] [fintype n] [fintype o] {α : Type v} {β : Type u_7} [semiring α] [semiring β] (M : matrix m n α) (N : matrix n o α) (i : m) (j : o) (f : α →+* β) :
f ((M N) i j) = ((λ (i : m) (j : n), f (M i j)) λ (i : n) (j : o), f (N i j)) i j