Matrices with a single non-zero element. #

This file provides Matrix.single. The matrix Matrix.single i j c has c at position (i, j), and zeroes elsewhere.

source

def Matrix.single {m : Type u_2} {n : Type u_3} {α : Type u_7} [DecidableEq m] [DecidableEq n] [Zero α] (i : m) (j : n) (a : α) :

Matrix m n α

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

Equations

Matrix.single i j a = Matrix.of fun (i' : m) (j' : n) => if i = i' ∧ j = j' then a else 0

Instances For

source

@[deprecated Matrix.single (since := "2025-05-05")]

def Matrix.stdBasisMatrix {m : Type u_2} {n : Type u_3} {α : Type u_7} [DecidableEq m] [DecidableEq n] [Zero α] (i : m) (j : n) (a : α) :

Matrix m n α

Alias of Matrix.single.

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

Equations

@Matrix.stdBasisMatrix = @Matrix.single

Instances For

source

theorem Matrix.single_eq_of_single_single {m : Type u_2} {n : Type u_3} {α : Type u_7} [DecidableEq m] [DecidableEq n] [Zero α] (i : m) (j : n) (a : α) :

single i j a = of (Pi.single i (Pi.single j a))

See also single_eq_updateRow_zero and single_eq_updateCol_zero.

source

@[deprecated Matrix.single_eq_of_single_single (since := "2025-05-05")]

theorem Matrix.stdBasisMatrix_eq_of_single_single {m : Type u_2} {n : Type u_3} {α : Type u_7} [DecidableEq m] [DecidableEq n] [Zero α] (i : m) (j : n) (a : α) :

single i j a = of (Pi.single i (Pi.single j a))

Alias of Matrix.single_eq_of_single_single.

See also single_eq_updateRow_zero and single_eq_updateCol_zero.

source

@[simp]

theorem Matrix.of_symm_single {m : Type u_2} {n : Type u_3} {α : Type u_7} [DecidableEq m] [DecidableEq n] [Zero α] (i : m) (j : n) (a : α) :

of.symm (single i j a) = Pi.single i (Pi.single j a)

source

@[deprecated Matrix.of_symm_single (since := "2025-05-05")]

theorem Matrix.of_symm_stdBasisMatrix {m : Type u_2} {n : Type u_3} {α : Type u_7} [DecidableEq m] [DecidableEq n] [Zero α] (i : m) (j : n) (a : α) :

of.symm (single i j a) = Pi.single i (Pi.single j a)

Alias of Matrix.of_symm_single.

source

@[simp]

theorem Matrix.smul_single {m : Type u_2} {n : Type u_3} {R : Type u_5} {α : Type u_7} [DecidableEq m] [DecidableEq n] [Zero α] [SMulZeroClass R α] (r : R) (i : m) (j : n) (a : α) :

r • single i j a = single i j (r • a)

source

@[deprecated Matrix.smul_single (since := "2025-05-05")]

theorem Matrix.smul_stdBasisMatrix {m : Type u_2} {n : Type u_3} {R : Type u_5} {α : Type u_7} [DecidableEq m] [DecidableEq n] [Zero α] [SMulZeroClass R α] (r : R) (i : m) (j : n) (a : α) :

r • single i j a = single i j (r • a)

Alias of Matrix.smul_single.

source

@[simp]

theorem Matrix.single_zero {m : Type u_2} {n : Type u_3} {α : Type u_7} [DecidableEq m] [DecidableEq n] [Zero α] (i : m) (j : n) :

single i j 0 = 0

source

@[deprecated Matrix.single_zero (since := "2025-05-05")]

theorem Matrix.stdBasisMatrix_zero {m : Type u_2} {n : Type u_3} {α : Type u_7} [DecidableEq m] [DecidableEq n] [Zero α] (i : m) (j : n) :

single i j 0 = 0

Alias of Matrix.single_zero.

source

@[simp]

theorem Matrix.transpose_single {m : Type u_2} {n : Type u_3} {α : Type u_7} [DecidableEq m] [DecidableEq n] [Zero α] (i : m) (j : n) (a : α) :

(single i j a).transpose = single j i a

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@[deprecated Matrix.transpose_single (since := "2025-05-05")]

theorem Matrix.transpose_stdBasisMatrix {m : Type u_2} {n : Type u_3} {α : Type u_7} [DecidableEq m] [DecidableEq n] [Zero α] (i : m) (j : n) (a : α) :

(single i j a).transpose = single j i a

Alias of Matrix.transpose_single.

source

@[simp]

theorem Matrix.map_single {m : Type u_2} {n : Type u_3} {α : Type u_7} [DecidableEq m] [DecidableEq n] [Zero α] (i : m) (j : n) (a : α) {β : Type u_10} [Zero β] {F : Type u_11} [FunLike F α β] [ZeroHomClass F α β] (f : F) :

(single i j a).map ⇑f = single i j (f a)

source

@[deprecated Matrix.map_single (since := "2025-05-05")]

theorem Matrix.map_stdBasisMatrix {m : Type u_2} {n : Type u_3} {α : Type u_7} [DecidableEq m] [DecidableEq n] [Zero α] (i : m) (j : n) (a : α) {β : Type u_10} [Zero β] {F : Type u_11} [FunLike F α β] [ZeroHomClass F α β] (f : F) :

(single i j a).map ⇑f = single i j (f a)

Alias of Matrix.map_single.

source

theorem Matrix.single_mem_matrix {m : Type u_2} {n : Type u_3} {α : Type u_7} [DecidableEq m] [DecidableEq n] [Zero α] {S : Set α} (hS : 0 ∈ S) {i : m} {j : n} {a : α} :

single i j a ∈ S.matrix ↔ a ∈ S

source

theorem Matrix.single_add {m : Type u_2} {n : Type u_3} {α : Type u_7} [DecidableEq m] [DecidableEq n] [AddZeroClass α] (i : m) (j : n) (a b : α) :

single i j (a + b) = single i j a + single i j b

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@[deprecated Matrix.single_add (since := "2025-05-05")]

theorem Matrix.stdBasisMatrix_add {m : Type u_2} {n : Type u_3} {α : Type u_7} [DecidableEq m] [DecidableEq n] [AddZeroClass α] (i : m) (j : n) (a b : α) :

single i j (a + b) = single i j a + single i j b

Alias of Matrix.single_add.

source

theorem Matrix.single_mulVec {m : Type u_2} {n : Type u_3} {α : Type u_7} [DecidableEq m] [DecidableEq n] [NonUnitalNonAssocSemiring α] [Fintype m] (i : n) (j : m) (c : α) (x : m → α) :

(single i j c).mulVec x = Function.update 0 i (c * x j)

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@[deprecated Matrix.single_mulVec (since := "2025-05-05")]

theorem Matrix.mulVec_stdBasisMatrix {m : Type u_2} {n : Type u_3} {α : Type u_7} [DecidableEq m] [DecidableEq n] [NonUnitalNonAssocSemiring α] [Fintype m] (i : n) (j : m) (c : α) (x : m → α) :

(single i j c).mulVec x = Function.update 0 i (c * x j)

Alias of Matrix.single_mulVec.

source

theorem Matrix.matrix_eq_sum_single {m : Type u_2} {n : Type u_3} {α : Type u_7} [DecidableEq m] [DecidableEq n] [AddCommMonoid α] [Fintype m] [Fintype n] (x : Matrix m n α) :

x = ∑ i : m, ∑ j : n, single i j (x i j)

source

@[deprecated Matrix.matrix_eq_sum_single (since := "2025-05-05")]

theorem Matrix.matrix_eq_sum_stdBasisMatrix {m : Type u_2} {n : Type u_3} {α : Type u_7} [DecidableEq m] [DecidableEq n] [AddCommMonoid α] [Fintype m] [Fintype n] (x : Matrix m n α) :

x = ∑ i : m, ∑ j : n, single i j (x i j)

Alias of Matrix.matrix_eq_sum_single.

source

theorem Matrix.single_eq_single_vecMulVec_single {m : Type u_2} {n : Type u_3} {α : Type u_7} [DecidableEq m] [DecidableEq n] [MulZeroOneClass α] (i : m) (j : n) :

single i j 1 = vecMulVec (Pi.single i 1) (Pi.single j 1)

source

@[deprecated Matrix.single_eq_single_vecMulVec_single (since := "2025-05-05")]

theorem Matrix.stdBasisMatrix_eq_single_vecMulVec_single {m : Type u_2} {n : Type u_3} {α : Type u_7} [DecidableEq m] [DecidableEq n] [MulZeroOneClass α] (i : m) (j : n) :

single i j 1 = vecMulVec (Pi.single i 1) (Pi.single j 1)

Alias of Matrix.single_eq_single_vecMulVec_single.

source

theorem Matrix.induction_on' {m : Type u_2} {n : Type u_3} {α : Type u_7} [DecidableEq m] [DecidableEq n] [AddCommMonoid α] [Finite m] [Finite n] {P : Matrix m n α → Prop} (M : Matrix m n α) (h_zero : P 0) (h_add : ∀ (p q : Matrix m n α), P p → P q → P (p + q)) (h_std_basis : ∀ (i : m) (j : n) (x : α), P (single i j x)) :

P M

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theorem Matrix.induction_on {m : Type u_2} {n : Type u_3} {α : Type u_7} [DecidableEq m] [DecidableEq n] [AddCommMonoid α] [Finite m] [Finite n] [Nonempty m] [Nonempty n] {P : Matrix m n α → Prop} (M : Matrix m n α) (h_add : ∀ (p q : Matrix m n α), P p → P q → P (p + q)) (h_std_basis : ∀ (i : m) (j : n) (x : α), P (single i j x)) :

P M

source

def Matrix.singleAddMonoidHom {m : Type u_2} {n : Type u_3} {α : Type u_7} [DecidableEq m] [DecidableEq n] [AddCommMonoid α] (i : m) (j : n) :

α →+ Matrix m n α

Matrix.single as a bundled additive map.

Equations

Matrix.singleAddMonoidHom i j = { toFun := Matrix.single i j, map_zero' := ⋯, map_add' := ⋯ }

Instances For

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@[simp]

theorem Matrix.singleAddMonoidHom_apply {m : Type u_2} {n : Type u_3} {α : Type u_7} [DecidableEq m] [DecidableEq n] [AddCommMonoid α] (i : m) (j : n) (a : α) :

(singleAddMonoidHom i j) a = single i j a

source

@[deprecated Matrix.singleAddMonoidHom (since := "2025-05-05")]

def Matrix.stdBasisMatrixAddMonoidHom {m : Type u_2} {n : Type u_3} {α : Type u_7} [DecidableEq m] [DecidableEq n] [AddCommMonoid α] (i : m) (j : n) :

α →+ Matrix m n α

Alias of Matrix.singleAddMonoidHom.

Matrix.single as a bundled additive map.

Equations

@Matrix.stdBasisMatrixAddMonoidHom = @Matrix.singleAddMonoidHom

Instances For

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def Matrix.singleLinearMap {m : Type u_2} {n : Type u_3} (R : Type u_5) {α : Type u_7} [DecidableEq m] [DecidableEq n] [Semiring R] [AddCommMonoid α] [Module R α] (i : m) (j : n) :

α →ₗ[R] Matrix m n α

Matrix.single as a bundled linear map.

Equations

Matrix.singleLinearMap R i j = { toFun := (↑(Matrix.singleAddMonoidHom i j)).toFun, map_add' := ⋯, map_smul' := ⋯ }

Instances For

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@[simp]

theorem Matrix.singleLinearMap_apply {m : Type u_2} {n : Type u_3} (R : Type u_5) {α : Type u_7} [DecidableEq m] [DecidableEq n] [Semiring R] [AddCommMonoid α] [Module R α] (i : m) (j : n) (a✝ : α) :

(singleLinearMap R i j) a✝ = single i j a✝

source

@[deprecated Matrix.singleLinearMap (since := "2025-05-05")]

def Matrix.stdBasisMatrixLinearMap {m : Type u_2} {n : Type u_3} (R : Type u_5) {α : Type u_7} [DecidableEq m] [DecidableEq n] [Semiring R] [AddCommMonoid α] [Module R α] (i : m) (j : n) :

α →ₗ[R] Matrix m n α

Alias of Matrix.singleLinearMap.

Matrix.single as a bundled linear map.

Equations

@Matrix.stdBasisMatrixLinearMap = @Matrix.singleLinearMap

Instances For

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theorem Matrix.ext_addMonoidHom {m : Type u_2} {n : Type u_3} {α : Type u_7} {β : Type u_8} [DecidableEq m] [DecidableEq n] [Finite m] [Finite n] [AddCommMonoid α] [AddCommMonoid β] ⦃f g : Matrix m n α →+ β⦄ (h : ∀ (i : m) (j : n), f.comp (singleAddMonoidHom i j) = g.comp (singleAddMonoidHom i j)) :

f = g

Additive maps from finite matrices are equal if they agree on the standard basis.

See note [partially-applied ext lemmas].

source

theorem Matrix.ext_addMonoidHom_iff {m : Type u_2} {n : Type u_3} {α : Type u_7} {β : Type u_8} [DecidableEq m] [DecidableEq n] [Finite m] [Finite n] [AddCommMonoid α] [AddCommMonoid β] {f g : Matrix m n α →+ β} :

f = g ↔ ∀ (i : m) (j : n), f.comp (singleAddMonoidHom i j) = g.comp (singleAddMonoidHom i j)

source

theorem Matrix.ext_linearMap {m : Type u_2} {n : Type u_3} (R : Type u_5) {α : Type u_7} {β : Type u_8} [DecidableEq m] [DecidableEq n] [Finite m] [Finite n] [Semiring R] [AddCommMonoid α] [AddCommMonoid β] [Module R α] [Module R β] ⦃f g : Matrix m n α →ₗ[R] β⦄ (h : ∀ (i : m) (j : n), f ∘ₗ singleLinearMap R i j = g ∘ₗ singleLinearMap R i j) :

f = g

Linear maps from finite matrices are equal if they agree on the standard basis.

See note [partially-applied ext lemmas].

source

theorem Matrix.ext_linearMap_iff {m : Type u_2} {n : Type u_3} {R : Type u_5} {α : Type u_7} {β : Type u_8} [DecidableEq m] [DecidableEq n] [Finite m] [Finite n] [Semiring R] [AddCommMonoid α] [AddCommMonoid β] [Module R α] [Module R β] {f g : Matrix m n α →ₗ[R] β} :

f = g ↔ ∀ (i : m) (j : n), f ∘ₗ singleLinearMap R i j = g ∘ₗ singleLinearMap R i j

source

def Matrix.liftLinear {m : Type u_2} {n : Type u_3} {R : Type u_5} (S : Type u_6) {α : Type u_7} {β : Type u_8} [DecidableEq m] [DecidableEq n] [Fintype m] [Fintype n] [Semiring R] [Semiring S] [AddCommMonoid α] [AddCommMonoid β] [Module R α] [Module R β] [Module S β] [SMulCommClass R S β] :

(m → n → α →ₗ[R] β) ≃ₗ[S] Matrix m n α →ₗ[R] β

Families of linear maps acting on each element are equivalent to linear maps from a matrix.

This can be thought of as the matrix version of LinearMap.lsum.

Equations

One or more equations did not get rendered due to their size.

Instances For

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theorem Matrix.liftLinear_apply {m : Type u_2} {n : Type u_3} {R : Type u_5} (S : Type u_6) {α : Type u_7} {β : Type u_8} [DecidableEq m] [DecidableEq n] [Fintype m] [Fintype n] [Semiring R] [Semiring S] [AddCommMonoid α] [AddCommMonoid β] [Module R α] [Module R β] [Module S β] [SMulCommClass R S β] (f : m → n → α →ₗ[R] β) (M : Matrix m n α) :

((liftLinear S) f) M = ∑ i : m, ∑ j : n, (f i j) (M i j)

source

@[simp]

theorem Matrix.liftLinear_single {m : Type u_2} {n : Type u_3} {R : Type u_5} (S : Type u_6) {α : Type u_7} {β : Type u_8} [DecidableEq m] [DecidableEq n] [Fintype m] [Fintype n] [Semiring R] [Semiring S] [AddCommMonoid α] [AddCommMonoid β] [Module R α] [Module R β] [Module S β] [SMulCommClass R S β] (f : m → n → α →ₗ[R] β) (i : m) (j : n) (a : α) :

((liftLinear S) f) (single i j a) = (f i j) a

source

@[deprecated Matrix.liftLinear_single (since := "2025-08-13")]

theorem Matrix.liftLinear_piSingle {m : Type u_2} {n : Type u_3} {R : Type u_5} (S : Type u_6) {α : Type u_7} {β : Type u_8} [DecidableEq m] [DecidableEq n] [Fintype m] [Fintype n] [Semiring R] [Semiring S] [AddCommMonoid α] [AddCommMonoid β] [Module R α] [Module R β] [Module S β] [SMulCommClass R S β] (f : m → n → α →ₗ[R] β) (i : m) (j : n) (a : α) :

((liftLinear S) f) (single i j a) = (f i j) a

Alias of Matrix.liftLinear_single.

source

@[simp]

theorem Matrix.liftLinear_comp_singleLinearMap {m : Type u_2} {n : Type u_3} {R : Type u_5} (S : Type u_6) {α : Type u_7} {β : Type u_8} [DecidableEq m] [DecidableEq n] [Fintype m] [Fintype n] [Semiring R] [Semiring S] [AddCommMonoid α] [AddCommMonoid β] [Module R α] [Module R β] [Module S β] [SMulCommClass R S β] (f : m → n → α →ₗ[R] β) (i : m) (j : n) :

(liftLinear S) f ∘ₗ singleLinearMap R i j = f i j

source

@[simp]

theorem Matrix.liftLinear_singleLinearMap {m : Type u_2} {n : Type u_3} {R : Type u_5} (S : Type u_6) {α : Type u_7} [DecidableEq m] [DecidableEq n] [Fintype m] [Fintype n] [Semiring R] [Semiring S] [AddCommMonoid α] [Module R α] [Module S α] [SMulCommClass R S α] :

(liftLinear S) (singleLinearMap R) = LinearMap.id

source

@[simp]

theorem Matrix.single_apply_same {m : Type u_2} {n : Type u_3} {α : Type u_7} [DecidableEq m] [DecidableEq n] [Zero α] (i : m) (j : n) (c : α) :

single i j c i j = c

source

@[deprecated Matrix.single_apply_same (since := "2025-05-05")]

theorem Matrix.StdBasisMatrix.apply_same {m : Type u_2} {n : Type u_3} {α : Type u_7} [DecidableEq m] [DecidableEq n] [Zero α] (i : m) (j : n) (c : α) :

single i j c i j = c

Alias of Matrix.single_apply_same.

source

@[simp]

theorem Matrix.single_apply_of_ne {m : Type u_2} {n : Type u_3} {α : Type u_7} [DecidableEq m] [DecidableEq n] [Zero α] (i : m) (j : n) (c : α) (i' : m) (j' : n) (h : ¬(i = i' ∧ j = j')) :

single i j c i' j' = 0

source

@[deprecated Matrix.single_apply_of_ne (since := "2025-05-05")]

theorem Matrix.StdBasisMatrix.apply_of_ne {m : Type u_2} {n : Type u_3} {α : Type u_7} [DecidableEq m] [DecidableEq n] [Zero α] (i : m) (j : n) (c : α) (i' : m) (j' : n) (h : ¬(i = i' ∧ j = j')) :

single i j c i' j' = 0

Alias of Matrix.single_apply_of_ne.

source

theorem Matrix.single_apply_of_row_ne {m : Type u_2} {n : Type u_3} {α : Type u_7} [DecidableEq m] [DecidableEq n] [Zero α] {i i' : m} (hi : i ≠ i') (j j' : n) (a : α) :

single i j a i' j' = 0

source

@[deprecated Matrix.single_apply_of_row_ne (since := "2025-05-05")]

theorem Matrix.StdBasisMatrix.apply_of_row_ne {m : Type u_2} {n : Type u_3} {α : Type u_7} [DecidableEq m] [DecidableEq n] [Zero α] {i i' : m} (hi : i ≠ i') (j j' : n) (a : α) :

single i j a i' j' = 0

Alias of Matrix.single_apply_of_row_ne.

source

theorem Matrix.single_apply_of_col_ne {m : Type u_2} {n : Type u_3} {α : Type u_7} [DecidableEq m] [DecidableEq n] [Zero α] (i i' : m) {j j' : n} (hj : j ≠ j') (a : α) :

single i j a i' j' = 0

source

@[deprecated Matrix.single_apply_of_col_ne (since := "2025-05-05")]

theorem Matrix.StdBasisMatrix.apply_of_col_ne {m : Type u_2} {n : Type u_3} {α : Type u_7} [DecidableEq m] [DecidableEq n] [Zero α] (i i' : m) {j j' : n} (hj : j ≠ j') (a : α) :

single i j a i' j' = 0

Alias of Matrix.single_apply_of_col_ne.

source

@[simp]

theorem Matrix.diag_single_of_ne {n : Type u_3} {α : Type u_7} [DecidableEq n] [Zero α] (i j : n) (c : α) (h : i ≠ j) :

(single i j c).diag = 0

source

@[deprecated Matrix.diag_single_of_ne (since := "2025-05-05")]

theorem Matrix.StdBasisMatrix.diag_zero {n : Type u_3} {α : Type u_7} [DecidableEq n] [Zero α] (i j : n) (c : α) (h : i ≠ j) :

(single i j c).diag = 0

Alias of Matrix.diag_single_of_ne.

source

@[simp]

theorem Matrix.diag_single_same {n : Type u_3} {α : Type u_7} [DecidableEq n] [Zero α] (i : n) (c : α) :

(single i i c).diag = Pi.single i c

source

@[deprecated Matrix.diag_single_same (since := "2025-05-05")]

theorem Matrix.StdBasisMatrix.diag_same {n : Type u_3} {α : Type u_7} [DecidableEq n] [Zero α] (i : n) (c : α) :

(single i i c).diag = Pi.single i c

Alias of Matrix.diag_single_same.

source

@[simp]

theorem Matrix.single_mul_apply_same {l : Type u_1} {m : Type u_2} {n : Type u_3} {α : Type u_7} [DecidableEq l] [DecidableEq m] [Fintype m] [NonUnitalNonAssocSemiring α] (c : α) (i : l) (j : m) (b : n) (M : Matrix m n α) :

(single i j c * M) i b = c * M j b

source

@[deprecated Matrix.single_mul_apply_same (since := "2025-05-05")]

theorem Matrix.StdBasisMatrix.mul_left_apply_same {l : Type u_1} {m : Type u_2} {n : Type u_3} {α : Type u_7} [DecidableEq l] [DecidableEq m] [Fintype m] [NonUnitalNonAssocSemiring α] (c : α) (i : l) (j : m) (b : n) (M : Matrix m n α) :

(single i j c * M) i b = c * M j b

Alias of Matrix.single_mul_apply_same.

source

@[simp]

theorem Matrix.mul_single_apply_same {l : Type u_1} {m : Type u_2} {n : Type u_3} {α : Type u_7} [DecidableEq m] [DecidableEq n] [Fintype m] [NonUnitalNonAssocSemiring α] (c : α) (i : m) (j : n) (a : l) (M : Matrix l m α) :

(M * single i j c) a j = M a i * c

source

@[deprecated Matrix.mul_single_apply_same (since := "2025-05-05")]

theorem Matrix.StdBasisMatrix.mul_right_apply_same {l : Type u_1} {m : Type u_2} {n : Type u_3} {α : Type u_7} [DecidableEq m] [DecidableEq n] [Fintype m] [NonUnitalNonAssocSemiring α] (c : α) (i : m) (j : n) (a : l) (M : Matrix l m α) :

(M * single i j c) a j = M a i * c

Alias of Matrix.mul_single_apply_same.

source

@[simp]

theorem Matrix.single_mul_apply_of_ne {l : Type u_1} {m : Type u_2} {n : Type u_3} {α : Type u_7} [DecidableEq l] [DecidableEq m] [Fintype m] [NonUnitalNonAssocSemiring α] (c : α) (i : l) (j : m) (a : l) (b : n) (h : a ≠ i) (M : Matrix m n α) :

(single i j c * M) a b = 0

source

@[deprecated Matrix.single_mul_apply_of_ne (since := "2025-05-05")]

theorem Matrix.StdBasisMatrix.mul_left_apply_of_ne {l : Type u_1} {m : Type u_2} {n : Type u_3} {α : Type u_7} [DecidableEq l] [DecidableEq m] [Fintype m] [NonUnitalNonAssocSemiring α] (c : α) (i : l) (j : m) (a : l) (b : n) (h : a ≠ i) (M : Matrix m n α) :

(single i j c * M) a b = 0

Alias of Matrix.single_mul_apply_of_ne.

source

@[simp]

theorem Matrix.mul_single_apply_of_ne {l : Type u_1} {m : Type u_2} {n : Type u_3} {α : Type u_7} [DecidableEq m] [DecidableEq n] [Fintype m] [NonUnitalNonAssocSemiring α] (c : α) (i : m) (j : n) (a : l) (b : n) (hbj : b ≠ j) (M : Matrix l m α) :

(M * single i j c) a b = 0

source

@[deprecated Matrix.mul_single_apply_of_ne (since := "2025-05-05")]

theorem Matrix.StdBasisMatrix.mul_right_apply_of_ne {l : Type u_1} {m : Type u_2} {n : Type u_3} {α : Type u_7} [DecidableEq m] [DecidableEq n] [Fintype m] [NonUnitalNonAssocSemiring α] (c : α) (i : m) (j : n) (a : l) (b : n) (hbj : b ≠ j) (M : Matrix l m α) :

(M * single i j c) a b = 0

Alias of Matrix.mul_single_apply_of_ne.

source

@[simp]

theorem Matrix.single_mul_single_same {l : Type u_1} {m : Type u_2} {n : Type u_3} {α : Type u_7} [DecidableEq l] [DecidableEq m] [DecidableEq n] [Fintype m] [NonUnitalNonAssocSemiring α] (c : α) (i : l) (j : m) (k : n) (d : α) :

single i j c * single j k d = single i k (c * d)

source

@[deprecated Matrix.single_mul_single_same (since := "2025-05-05")]

theorem Matrix.StdBasisMatrix.mul_same {l : Type u_1} {m : Type u_2} {n : Type u_3} {α : Type u_7} [DecidableEq l] [DecidableEq m] [DecidableEq n] [Fintype m] [NonUnitalNonAssocSemiring α] (c : α) (i : l) (j : m) (k : n) (d : α) :

single i j c * single j k d = single i k (c * d)

Alias of Matrix.single_mul_single_same.

source

@[simp]

theorem Matrix.single_mul_mul_single {l : Type u_1} {m : Type u_2} {n : Type u_3} {o : Type u_4} {α : Type u_7} [DecidableEq l] [DecidableEq m] [DecidableEq n] [DecidableEq o] [Fintype m] [NonUnitalNonAssocSemiring α] [Fintype n] (i : l) (i' : m) (j' : n) (j : o) (a : α) (x : Matrix m n α) (b : α) :

single i i' a * x * single j' j b = single i j (a * x i' j' * b)

source

@[deprecated Matrix.single_mul_mul_single (since := "2025-05-05")]

theorem Matrix.StdBasisMatrix.stdBasisMatrix_mul_mul_stdBasisMatrix {l : Type u_1} {m : Type u_2} {n : Type u_3} {o : Type u_4} {α : Type u_7} [DecidableEq l] [DecidableEq m] [DecidableEq n] [DecidableEq o] [Fintype m] [NonUnitalNonAssocSemiring α] [Fintype n] (i : l) (i' : m) (j' : n) (j : o) (a : α) (x : Matrix m n α) (b : α) :

single i i' a * x * single j' j b = single i j (a * x i' j' * b)

Alias of Matrix.single_mul_mul_single.

source

@[simp]

theorem Matrix.single_mul_single_of_ne {l : Type u_1} {m : Type u_2} {n : Type u_3} {α : Type u_7} [DecidableEq l] [DecidableEq m] [DecidableEq n] [Fintype m] [NonUnitalNonAssocSemiring α] (c : α) (i : l) (j k : m) {l✝ : n} (h : j ≠ k) (d : α) :

single i j c * single k l✝ d = 0

source

@[deprecated Matrix.single_mul_single_of_ne (since := "2025-05-05")]

theorem Matrix.StdBasisMatrix.mul_of_ne {l : Type u_1} {m : Type u_2} {n : Type u_3} {α : Type u_7} [DecidableEq l] [DecidableEq m] [DecidableEq n] [Fintype m] [NonUnitalNonAssocSemiring α] (c : α) (i : l) (j k : m) {l✝ : n} (h : j ≠ k) (d : α) :

single i j c * single k l✝ d = 0

Alias of Matrix.single_mul_single_of_ne.

source

theorem Matrix.row_eq_zero_of_commute_single {n : Type u_3} {α : Type u_7} [DecidableEq n] [Fintype n] [Semiring α] {i j k : n} {M : Matrix n n α} (hM : Commute (single i j 1) M) (hkj : k ≠ j) :

M j k = 0

source

@[deprecated Matrix.row_eq_zero_of_commute_single (since := "2025-05-05")]

theorem Matrix.row_eq_zero_of_commute_stdBasisMatrix {n : Type u_3} {α : Type u_7} [DecidableEq n] [Fintype n] [Semiring α] {i j k : n} {M : Matrix n n α} (hM : Commute (single i j 1) M) (hkj : k ≠ j) :

M j k = 0

Alias of Matrix.row_eq_zero_of_commute_single.

source

theorem Matrix.col_eq_zero_of_commute_single {n : Type u_3} {α : Type u_7} [DecidableEq n] [Fintype n] [Semiring α] {i j k : n} {M : Matrix n n α} (hM : Commute (single i j 1) M) (hki : k ≠ i) :

M k i = 0

source

@[deprecated Matrix.col_eq_zero_of_commute_single (since := "2025-05-05")]

theorem Matrix.col_eq_zero_of_commute_stdBasisMatrix {n : Type u_3} {α : Type u_7} [DecidableEq n] [Fintype n] [Semiring α] {i j k : n} {M : Matrix n n α} (hM : Commute (single i j 1) M) (hki : k ≠ i) :

M k i = 0

Alias of Matrix.col_eq_zero_of_commute_single.

source

theorem Matrix.diag_eq_of_commute_single {n : Type u_3} {α : Type u_7} [DecidableEq n] [Fintype n] [Semiring α] {i j : n} {M : Matrix n n α} (hM : Commute (single i j 1) M) :

M i i = M j j

source

@[deprecated Matrix.diag_eq_of_commute_single (since := "2025-05-05")]

theorem Matrix.diag_eq_of_commute_stdBasisMatrix {n : Type u_3} {α : Type u_7} [DecidableEq n] [Fintype n] [Semiring α] {i j : n} {M : Matrix n n α} (hM : Commute (single i j 1) M) :

M i i = M j j

Alias of Matrix.diag_eq_of_commute_single.

source

theorem Matrix.mem_range_scalar_of_commute_single {n : Type u_3} {α : Type u_7} [DecidableEq n] [Fintype n] [Semiring α] {M : Matrix n n α} (hM : Pairwise fun (i j : n) => Commute (single i j 1) M) :

M ∈ Set.range ⇑(scalar n)

M is a scalar matrix if it commutes with every non-diagonal single.

source

@[deprecated Matrix.mem_range_scalar_of_commute_single (since := "2025-05-05")]

theorem Matrix.mem_range_scalar_of_commute_stdBasisMatrix {n : Type u_3} {α : Type u_7} [DecidableEq n] [Fintype n] [Semiring α] {M : Matrix n n α} (hM : Pairwise fun (i j : n) => Commute (single i j 1) M) :

M ∈ Set.range ⇑(scalar n)

Alias of Matrix.mem_range_scalar_of_commute_single.

M is a scalar matrix if it commutes with every non-diagonal single.

source

theorem Matrix.mem_range_scalar_iff_commute_single {n : Type u_3} {α : Type u_7} [DecidableEq n] [Fintype n] [Semiring α] {M : Matrix n n α} :

M ∈ Set.range ⇑(scalar n) ↔ ∀ (i j : n), i ≠ j → Commute (single i j 1) M

source

@[deprecated Matrix.mem_range_scalar_iff_commute_single (since := "2025-05-05")]

theorem Matrix.mem_range_scalar_iff_commute_stdBasisMatrix {n : Type u_3} {α : Type u_7} [DecidableEq n] [Fintype n] [Semiring α] {M : Matrix n n α} :

M ∈ Set.range ⇑(scalar n) ↔ ∀ (i j : n), i ≠ j → Commute (single i j 1) M

Alias of Matrix.mem_range_scalar_iff_commute_single.

source

theorem Matrix.mem_range_scalar_iff_commute_single' {n : Type u_3} {α : Type u_7} [DecidableEq n] [Fintype n] [Semiring α] {M : Matrix n n α} :

M ∈ Set.range ⇑(scalar n) ↔ ∀ (i j : n), Commute (single i j 1) M

M is a scalar matrix if and only if it commutes with every single.

source

@[deprecated Matrix.mem_range_scalar_iff_commute_single' (since := "2025-05-05")]

theorem Matrix.mem_range_scalar_iff_commute_stdBasisMatrix' {n : Type u_3} {α : Type u_7} [DecidableEq n] [Fintype n] [Semiring α] {M : Matrix n n α} :

M ∈ Set.range ⇑(scalar n) ↔ ∀ (i j : n), Commute (single i j 1) M

Alias of Matrix.mem_range_scalar_iff_commute_single'.

M is a scalar matrix if and only if it commutes with every single.

source

theorem Matrix.center_eq_scalar_image {n : Type u_3} {α : Type u_7} [DecidableEq n] [Fintype n] [Semiring α] :

Set.center (Matrix n n α) = ⇑(scalar n) '' Set.center α

The center of Matrix n n α is equal to the image of the center of α under scalar n.

source

theorem Matrix.submonoidCenter_eq_scalar_map {n : Type u_3} {α : Type u_7} [DecidableEq n] [Fintype n] [Semiring α] :

Submonoid.center (Matrix n n α) = Submonoid.map (scalar n) (Submonoid.center α)

source

theorem Matrix.subsemigroupCenter_eq_scalar_map {n : Type u_3} {α : Type u_7} [DecidableEq n] [Fintype n] [Semiring α] :

Subsemigroup.center (Matrix n n α) = Subsemigroup.map (↑↑(scalar n)) (Subsemigroup.center α)

source

theorem Matrix.subsemiringCenter_eq_scalar_map {n : Type u_3} {α : Type u_7} [DecidableEq n] [Fintype n] [Semiring α] :

Subsemiring.center (Matrix n n α) = Subsemiring.map (scalar n) (Subsemiring.center α)

source

theorem Matrix.subringCenter_eq_scalar_map {n : Type u_3} (R : Type u_5) [DecidableEq n] [Fintype n] [Ring R] :

Subring.center (Matrix n n R) = Subring.map (scalar n) (Subring.center R)

source

@[simp]

theorem Matrix.center_eq_range {n : Type u_3} (R : Type u_5) [DecidableEq n] [Fintype n] [CommSemiring R] :

Set.center (Matrix n n R) = Set.range ⇑(scalar n)

For a commutative semiring R, the center of Matrix n n R is the range of scalar n (i.e., the span of {1}).

Documentation

Mathlib.Data.Matrix.Basis

Matrices with a single non-zero element. #