mathlib documentation

data.matrix.block

Block Matrices #

Main definitions #

def matrix.from_blocks {l : Type u_1} {m : Type u_2} {n : Type u_3} {o : Type u_4} {α : Type u_12} (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} {α : Type u_12} (A : matrix n l α) (B : matrix n m α) (C : matrix o l α) (D : matrix o m α) (i : n) (j : l) :
matrix.from_blocks A 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} {α : Type u_12} (A : matrix n l α) (B : matrix n m α) (C : matrix o l α) (D : matrix o m α) (i : n) (j : m) :
matrix.from_blocks A 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} {α : Type u_12} (A : matrix n l α) (B : matrix n m α) (C : matrix o l α) (D : matrix o m α) (i : o) (j : l) :
matrix.from_blocks A 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} {α : Type u_12} (A : matrix n l α) (B : matrix n m α) (C : matrix o l α) (D : matrix o m α) (i : o) (j : m) :
matrix.from_blocks A 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} {α : Type u_12} (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} {α : Type u_12} (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} {α : Type u_12} (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} {α : Type u_12} (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} {α : Type u_12} (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} {α : Type u_12} (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} {α : Type u_12} (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} {α : Type u_12} (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} {α : Type u_12} (A : matrix n l α) (B : matrix n m α) (C : matrix o l α) (D : matrix o m α) :
theorem matrix.from_blocks_map {l : Type u_1} {m : Type u_2} {n : Type u_3} {o : Type u_4} {α : Type u_12} {β : Type u_13} (A : matrix n l α) (B : matrix n m α) (C : matrix o l α) (D : matrix o m α) (f : α → β) :
(matrix.from_blocks A B C D).map f = matrix.from_blocks (A.map f) (B.map f) (C.map f) (D.map f)
theorem matrix.from_blocks_transpose {l : Type u_1} {m : Type u_2} {n : Type u_3} {o : Type u_4} {α : Type u_12} (A : matrix n l α) (B : matrix n m α) (C : matrix o l α) (D : matrix o m α) :
theorem matrix.from_blocks_conj_transpose {l : Type u_1} {m : Type u_2} {n : Type u_3} {o : Type u_4} {α : Type u_12} [has_star α] (A : matrix n l α) (B : matrix n m α) (C : matrix o l α) (D : matrix o m α) :
def matrix.is_two_block_diagonal {l : Type u_1} {m : Type u_2} {n : Type u_3} {o : Type u_4} {α : Type u_12} [has_zero α] (A : matrix (n o) (l m) α) :
Prop

A 2x2 block matrix is block diagonal if the blocks outside of the diagonal vanish

Equations
def matrix.to_block {m : Type u_2} {n : Type u_3} {α : Type u_12} (M : matrix m n α) (p : m → Prop) (q : n → Prop) :
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} {α : Type u_12} (M : matrix m n α) (p : m → Prop) (q : n → Prop) (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} {α : Type u_12} (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} {α : Type u_12} (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} {α : Type u_12} (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} {α : Type u_12} (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} {α : Type u_12} (M : matrix m m α) (p : m → Prop) :
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} {α : Type u_12} (M : matrix m m α) (p : m → Prop) :
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} {R : Type u_10} {α : Type u_12} [has_scalar R α] (x : R) (A : matrix n l α) (B : matrix n m α) (C : matrix o l α) (D : matrix o m α) :
x matrix.from_blocks A B C D = matrix.from_blocks (x A) (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} {α : Type u_12} [has_add 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) :
matrix.from_blocks A B C D + matrix.from_blocks A' B' C' D' = matrix.from_blocks (A + A') (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} {p : Type u_5} {q : Type u_6} {α : Type u_12} [fintype l] [fintype m] [non_unital_non_assoc_semiring α] (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) :
(matrix.from_blocks A B C D).mul (matrix.from_blocks A' B' C' D') = matrix.from_blocks (A.mul A' + B.mul C') (A.mul B' + B.mul D') (C.mul A' + D.mul C') (C.mul B' + D.mul D')
theorem matrix.from_blocks_mul_vec {l : Type u_1} {m : Type u_2} {n : Type u_3} {o : Type u_4} {α : Type u_12} [fintype l] [fintype m] [non_unital_non_assoc_semiring α] (A : matrix n l α) (B : matrix n m α) (C : matrix o l α) (D : matrix o m α) (x : l m → α) :
theorem matrix.vec_mul_from_blocks {l : Type u_1} {m : Type u_2} {n : Type u_3} {o : Type u_4} {α : Type u_12} [fintype n] [fintype o] [non_unital_non_assoc_semiring α] (A : matrix n l α) (B : matrix n m α) (C : matrix o l α) (D : matrix o m α) (x : n o → α) :
@[simp]
theorem matrix.from_blocks_diagonal {l : Type u_1} {m : Type u_2} {α : Type u_12} [decidable_eq l] [decidable_eq m] [has_zero α] (d₁ : l → α) (d₂ : m → α) :
@[simp]
theorem matrix.from_blocks_one {l : Type u_1} {m : Type u_2} {α : Type u_12} [decidable_eq l] [decidable_eq m] [has_zero α] [has_one α] :
def matrix.block_diagonal {m : Type u_2} {n : Type u_3} {o : Type u_4} {α : Type u_12} [decidable_eq o] [has_zero α] (M : o → matrix m n 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} {α : Type u_12} [decidable_eq o] [has_zero α] (M : o → matrix m n 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} {α : Type u_12} [decidable_eq o] [has_zero α] (M : o → matrix m n 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} {α : Type u_12} [decidable_eq o] [has_zero α] (M : o → matrix m n α) (i : m) (j : n) {k k' : o} (h : k k') :
matrix.block_diagonal M (i, k) (j, k') = 0
theorem matrix.block_diagonal_map {m : Type u_2} {n : Type u_3} {o : Type u_4} {α : Type u_12} {β : Type u_13} [decidable_eq o] [has_zero α] [has_zero β] (M : o → matrix m n α) (f : α → β) (hf : f 0 = 0) :
@[simp]
theorem matrix.block_diagonal_transpose {m : Type u_2} {n : Type u_3} {o : Type u_4} {α : Type u_12} [decidable_eq o] [has_zero α] (M : o → matrix m n α) :
@[simp]
theorem matrix.block_diagonal_conj_transpose {m : Type u_2} {n : Type u_3} {o : Type u_4} [decidable_eq o] {α : Type u_1} [add_monoid α] [star_add_monoid α] (M : o → matrix m n α) :
@[simp]
theorem matrix.block_diagonal_zero {m : Type u_2} {n : Type u_3} {o : Type u_4} {α : Type u_12} [decidable_eq o] [has_zero α] :
@[simp]
theorem matrix.block_diagonal_diagonal {m : Type u_2} {o : Type u_4} {α : Type u_12} [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} {α : Type u_12} [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} {α : Type u_12} [decidable_eq o] [add_zero_class α] (M N : o → matrix m n α) :
@[simp]
theorem matrix.block_diagonal_add_monoid_hom_apply (m : Type u_2) (n : Type u_3) (o : Type u_4) (α : Type u_12) [decidable_eq o] [add_zero_class α] (M : o → matrix m n α) :
def matrix.block_diagonal_add_monoid_hom (m : Type u_2) (n : Type u_3) (o : Type u_4) (α : Type u_12) [decidable_eq o] [add_zero_class α] :
(o → matrix m n α) →+ matrix (m × o) (n × o) α

matrix.block_diagonal as an add_monoid_hom.

Equations
@[simp]
theorem matrix.block_diagonal_neg {m : Type u_2} {n : Type u_3} {o : Type u_4} {α : Type u_12} [decidable_eq o] [add_group α] (M : o → matrix m n α) :
@[simp]
theorem matrix.block_diagonal_sub {m : Type u_2} {n : Type u_3} {o : Type u_4} {α : Type u_12} [decidable_eq o] [add_group α] (M N : o → matrix m n α) :
@[simp]
theorem matrix.block_diagonal_mul {m : Type u_2} {n : Type u_3} {o : Type u_4} {p : Type u_5} {α : Type u_12} [decidable_eq o] [fintype n] [fintype o] [non_unital_non_assoc_semiring α] (M : o → matrix m n α) (N : o → matrix n p α) :
@[simp]
theorem matrix.block_diagonal_ring_hom_apply (m : Type u_2) (o : Type u_4) (α : Type u_12) [decidable_eq o] [decidable_eq m] [fintype o] [fintype m] [non_assoc_semiring α] (M : o → matrix m m α) :
def matrix.block_diagonal_ring_hom (m : Type u_2) (o : Type u_4) (α : Type u_12) [decidable_eq o] [decidable_eq m] [fintype o] [fintype m] [non_assoc_semiring α] :
(o → matrix m m α) →+* matrix (m × o) (m × o) α

matrix.block_diagonal as a ring_hom.

Equations
@[simp]
theorem matrix.block_diagonal_pow {m : Type u_2} {o : Type u_4} {α : Type u_12} [decidable_eq o] [decidable_eq m] [fintype o] [fintype m] [semiring α] (M : o → matrix m m α) (n : ) :
@[simp]
theorem matrix.block_diagonal_smul {m : Type u_2} {n : Type u_3} {o : Type u_4} {α : Type u_12} [decidable_eq o] {R : Type u_1} [monoid R] [add_monoid α] [distrib_mul_action R α] (x : R) (M : o → matrix m n α) :
def matrix.block_diag {m : Type u_2} {n : Type u_3} {o : Type u_4} {α : Type u_12} (M : matrix (m × o) (n × o) α) (k : o) :
matrix m n α

Extract a block from the diagonal of a block diagonal matrix.

This is the block form of matrix.diag, and the left-inverse of matrix.block_diagonal.

Equations
theorem matrix.block_diag_map {m : Type u_2} {n : Type u_3} {o : Type u_4} {α : Type u_12} {β : Type u_13} (M : matrix (m × o) (n × o) α) (f : α → β) :
(M.map f).block_diag = λ (k : o), (M.block_diag k).map f
@[simp]
theorem matrix.block_diag_transpose {m : Type u_2} {n : Type u_3} {o : Type u_4} {α : Type u_12} (M : matrix (m × o) (n × o) α) (k : o) :
@[simp]
theorem matrix.block_diag_conj_transpose {m : Type u_2} {n : Type u_3} {o : Type u_4} {α : Type u_1} [add_monoid α] [star_add_monoid α] (M : matrix (m × o) (n × o) α) (k : o) :
@[simp]
theorem matrix.block_diag_zero {m : Type u_2} {n : Type u_3} {o : Type u_4} {α : Type u_12} [has_zero α] :
@[simp]
theorem matrix.block_diag_diagonal {m : Type u_2} {o : Type u_4} {α : Type u_12} [has_zero α] [decidable_eq o] [decidable_eq m] (d : m × o → α) (k : o) :
(matrix.diagonal d).block_diag k = matrix.diagonal (λ (i : m), d (i, k))
@[simp]
theorem matrix.block_diag_block_diagonal {m : Type u_2} {n : Type u_3} {o : Type u_4} {α : Type u_12} [has_zero α] [decidable_eq o] (M : o → matrix m n α) :
@[simp]
theorem matrix.block_diag_one {m : Type u_2} {o : Type u_4} {α : Type u_12} [has_zero α] [decidable_eq o] [decidable_eq m] [has_one α] :
@[simp]
theorem matrix.block_diag_add {m : Type u_2} {n : Type u_3} {o : Type u_4} {α : Type u_12} [add_zero_class α] (M N : matrix (m × o) (n × o) α) :
def matrix.block_diag_add_monoid_hom (m : Type u_2) (n : Type u_3) (o : Type u_4) (α : Type u_12) [add_zero_class α] :
matrix (m × o) (n × o) α →+ o → matrix m n α

matrix.block_diag as an add_monoid_hom.

Equations
@[simp]
theorem matrix.block_diag_add_monoid_hom_apply (m : Type u_2) (n : Type u_3) (o : Type u_4) (α : Type u_12) [add_zero_class α] (M : matrix (m × o) (n × o) α) (k : o) :
@[simp]
theorem matrix.block_diag_neg {m : Type u_2} {n : Type u_3} {o : Type u_4} {α : Type u_12} [add_group α] (M : matrix (m × o) (n × o) α) :
@[simp]
theorem matrix.block_diag_sub {m : Type u_2} {n : Type u_3} {o : Type u_4} {α : Type u_12} [add_group α] (M N : matrix (m × o) (n × o) α) :
@[simp]
theorem matrix.block_diag_smul {m : Type u_2} {n : Type u_3} {o : Type u_4} {α : Type u_12} {R : Type u_1} [monoid R] [add_monoid α] [distrib_mul_action R α] (x : R) (M : matrix (m × o) (n × o) α) :
def matrix.block_diagonal' {o : Type u_4} {m' : o → Type u_7} {n' : o → Type u_8} {α : Type u_12} [decidable_eq o] [has_zero α] (M : Π (i : o), matrix (m' i) (n' i) 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} {α : Type u_12} [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} {α : Type u_12} [decidable_eq o] [has_zero α] (M : o → matrix m n α) :
theorem matrix.block_diagonal'_apply {o : Type u_4} {m' : o → Type u_7} {n' : o → Type u_8} {α : Type u_12} [decidable_eq o] [has_zero α] (M : Π (i : o), matrix (m' i) (n' i) 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} {m' : o → Type u_7} {n' : o → Type u_8} {α : Type u_12} [decidable_eq o] [has_zero α] (M : Π (i : o), matrix (m' i) (n' i) 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} {m' : o → Type u_7} {n' : o → Type u_8} {α : Type u_12} [decidable_eq o] [has_zero α] (M : Π (i : o), matrix (m' i) (n' i) α) {k k' : o} (i : m' k) (j : n' k') (h : k k') :
matrix.block_diagonal' M k, i⟩ k', j⟩ = 0
theorem matrix.block_diagonal'_map {o : Type u_4} {m' : o → Type u_7} {n' : o → Type u_8} {α : Type u_12} {β : Type u_13} [decidable_eq o] [has_zero α] [has_zero β] (M : Π (i : o), matrix (m' i) (n' i) α) (f : α → β) (hf : f 0 = 0) :
@[simp]
theorem matrix.block_diagonal'_transpose {o : Type u_4} {m' : o → Type u_7} {n' : o → Type u_8} {α : Type u_12} [decidable_eq o] [has_zero α] (M : Π (i : o), matrix (m' i) (n' i) α) :
@[simp]
theorem matrix.block_diagonal'_conj_transpose {o : Type u_4} {m' : o → Type u_7} {n' : o → Type u_8} [decidable_eq o] {α : Type u_1} [add_monoid α] [star_add_monoid α] (M : Π (i : o), matrix (m' i) (n' i) α) :
@[simp]
theorem matrix.block_diagonal'_zero {o : Type u_4} {m' : o → Type u_7} {n' : o → Type u_8} {α : Type u_12} [decidable_eq o] [has_zero α] :
@[simp]
theorem matrix.block_diagonal'_diagonal {o : Type u_4} {m' : o → Type u_7} {α : Type u_12} [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} {m' : o → Type u_7} {α : Type u_12} [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} {m' : o → Type u_7} {n' : o → Type u_8} {α : Type u_12} [decidable_eq o] [add_zero_class α] (M N : Π (i : o), matrix (m' i) (n' i) α) :
def matrix.block_diagonal'_add_monoid_hom {o : Type u_4} (m' : o → Type u_7) (n' : o → Type u_8) (α : Type u_12) [decidable_eq o] [add_zero_class α] :
(Π (i : o), matrix (m' i) (n' i) α) →+ matrix (Σ (i : o), m' i) (Σ (i : o), n' i) α

matrix.block_diagonal' as an add_monoid_hom.

Equations
@[simp]
theorem matrix.block_diagonal'_add_monoid_hom_apply {o : Type u_4} (m' : o → Type u_7) (n' : o → Type u_8) (α : Type u_12) [decidable_eq o] [add_zero_class α] (M : Π (i : o), matrix ((λ (i : o), m' i) i) ((λ (i : o), n' i) i) α) :
@[simp]
theorem matrix.block_diagonal'_neg {o : Type u_4} {m' : o → Type u_7} {n' : o → Type u_8} {α : Type u_12} [decidable_eq o] [add_group α] (M : Π (i : o), matrix (m' i) (n' i) α) :
@[simp]
theorem matrix.block_diagonal'_sub {o : Type u_4} {m' : o → Type u_7} {n' : o → Type u_8} {α : Type u_12} [decidable_eq o] [add_group α] (M N : Π (i : o), matrix (m' i) (n' i) α) :
@[simp]
theorem matrix.block_diagonal'_mul {o : Type u_4} {m' : o → Type u_7} {n' : o → Type u_8} {p' : o → Type u_9} {α : Type u_12} [decidable_eq o] [non_unital_non_assoc_semiring α] [Π (i : o), fintype (n' i)] [fintype o] (M : Π (i : o), matrix (m' i) (n' i) α) (N : Π (i : o), matrix (n' i) (p' i) α) :
def matrix.block_diagonal'_ring_hom {o : Type u_4} (m' : o → Type u_7) (α : Type u_12) [decidable_eq o] [Π (i : o), decidable_eq (m' i)] [fintype o] [Π (i : o), fintype (m' i)] [non_assoc_semiring α] :
(Π (i : o), matrix (m' i) (m' i) α) →+* matrix (Σ (i : o), m' i) (Σ (i : o), m' i) α

matrix.block_diagonal' as a ring_hom.

Equations
@[simp]
theorem matrix.block_diagonal'_ring_hom_apply {o : Type u_4} (m' : o → Type u_7) (α : Type u_12) [decidable_eq o] [Π (i : o), decidable_eq (m' i)] [fintype o] [Π (i : o), fintype (m' i)] [non_assoc_semiring α] (M : Π (i : o), matrix ((λ (i : o), m' i) i) ((λ (i : o), m' i) i) α) :
@[simp]
theorem matrix.block_diagonal'_pow {o : Type u_4} {m' : o → Type u_7} {α : Type u_12} [decidable_eq o] [Π (i : o), decidable_eq (m' i)] [fintype o] [Π (i : o), fintype (m' i)] [semiring α] (M : Π (i : o), matrix (m' i) (m' i) α) (n : ) :
@[simp]
theorem matrix.block_diagonal'_smul {o : Type u_4} {m' : o → Type u_7} {n' : o → Type u_8} {α : Type u_12} [decidable_eq o] {R : Type u_1} [semiring R] [add_comm_monoid α] [module R α] (x : R) (M : Π (i : o), matrix (m' i) (n' i) α) :
def matrix.block_diag' {o : Type u_4} {m' : o → Type u_7} {n' : o → Type u_8} {α : Type u_12} (M : matrix (Σ (i : o), m' i) (Σ (i : o), n' i) α) (k : o) :
matrix (m' k) (n' k) α

Extract a block from the diagonal of a block diagonal matrix.

This is the block form of matrix.diag, and the left-inverse of matrix.block_diagonal'.

Equations
theorem matrix.block_diag'_map {o : Type u_4} {m' : o → Type u_7} {n' : o → Type u_8} {α : Type u_12} {β : Type u_13} (M : matrix (Σ (i : o), m' i) (Σ (i : o), n' i) α) (f : α → β) :
(M.map f).block_diag' = λ (k : o), (M.block_diag' k).map f
@[simp]
theorem matrix.block_diag'_transpose {o : Type u_4} {m' : o → Type u_7} {n' : o → Type u_8} {α : Type u_12} (M : matrix (Σ (i : o), m' i) (Σ (i : o), n' i) α) (k : o) :
@[simp]
theorem matrix.block_diag'_conj_transpose {o : Type u_4} {m' : o → Type u_7} {n' : o → Type u_8} {α : Type u_1} [add_monoid α] [star_add_monoid α] (M : matrix (Σ (i : o), m' i) (Σ (i : o), n' i) α) (k : o) :
@[simp]
theorem matrix.block_diag'_zero {o : Type u_4} {m' : o → Type u_7} {n' : o → Type u_8} {α : Type u_12} [has_zero α] :
@[simp]
theorem matrix.block_diag'_diagonal {o : Type u_4} {m' : o → Type u_7} {α : Type u_12} [has_zero α] [decidable_eq o] [Π (i : o), decidable_eq (m' i)] (d : (Σ (i : o), m' i) → α) (k : o) :
(matrix.diagonal d).block_diag' k = matrix.diagonal (λ (i : m' k), d k, i⟩)
@[simp]
theorem matrix.block_diag'_block_diagonal' {o : Type u_4} {m' : o → Type u_7} {n' : o → Type u_8} {α : Type u_12} [has_zero α] [decidable_eq o] (M : Π (i : o), matrix (m' i) (n' i) α) :
@[simp]
theorem matrix.block_diag'_one {o : Type u_4} {m' : o → Type u_7} {α : Type u_12} [has_zero α] [decidable_eq o] [Π (i : o), decidable_eq (m' i)] [has_one α] :
@[simp]
theorem matrix.block_diag'_add {o : Type u_4} {m' : o → Type u_7} {n' : o → Type u_8} {α : Type u_12} [add_zero_class α] (M N : matrix (Σ (i : o), m' i) (Σ (i : o), n' i) α) :
@[simp]
theorem matrix.block_diag'_add_monoid_hom_apply {o : Type u_4} (m' : o → Type u_7) (n' : o → Type u_8) (α : Type u_12) [add_zero_class α] (M : matrix (Σ (i : o), (λ (i : o), m' i) i) (Σ (i : o), (λ (i : o), n' i) i) α) (k : o) :
def matrix.block_diag'_add_monoid_hom {o : Type u_4} (m' : o → Type u_7) (n' : o → Type u_8) (α : Type u_12) [add_zero_class α] :
matrix (Σ (i : o), m' i) (Σ (i : o), n' i) α →+ Π (i : o), matrix (m' i) (n' i) α

matrix.block_diag' as an add_monoid_hom.

Equations
@[simp]
theorem matrix.block_diag'_neg {o : Type u_4} {m' : o → Type u_7} {n' : o → Type u_8} {α : Type u_12} [add_group α] (M : matrix (Σ (i : o), m' i) (Σ (i : o), n' i) α) :
@[simp]
theorem matrix.block_diag'_sub {o : Type u_4} {m' : o → Type u_7} {n' : o → Type u_8} {α : Type u_12} [add_group α] (M N : matrix (Σ (i : o), m' i) (Σ (i : o), n' i) α) :
@[simp]
theorem matrix.block_diag'_smul {o : Type u_4} {m' : o → Type u_7} {n' : o → Type u_8} {α : Type u_12} {R : Type u_1} [monoid R] [add_monoid α] [distrib_mul_action R α] (x : R) (M : matrix (Σ (i : o), m' i) (Σ (i : o), n' i) α) :