Row and column matrices

This file provides results about row and column matrices.

Main definitions

• Matrix.replicateRow ι r : Matrix ι n α: the matrix where every row is the vector r : n → α
• Matrix.replicateCol ι c : Matrix m ι α: the matrix where every column is the vector c : m → α
• Matrix.updateRow M i r: update the ith row of M to r
• Matrix.updateCol M j c: update the jth column of M to c

def Matrix.replicateCol {m : Type u_2} {α : Type v} (ι : Type u_6) (w : m → α) : Matrix m ι α

Matrix.replicateCol ι u is the matrix with all columns equal to the vector u.

To get a column matrix with exactly one column, Matrix.replicateCol (Fin 1) u is the canonical choice.

Equations

• Matrix.replicateCol ι w = Matrix.of fun (x : m) (x_1 : ι) => w x

@[deprecated Matrix.replicateCol (since := "2025-03-20")]

def Matrix.col {m : Type u_2} {α : Type v} (ι : Type u_6) (w : m → α) : Matrix m ι α

Alias of Matrix.replicateCol.

---

Matrix.replicateCol ι u is the matrix with all columns equal to the vector u.

To get a column matrix with exactly one column, Matrix.replicateCol (Fin 1) u is the canonical choice.

Equations

• @Matrix.col = @Matrix.replicateCol

@[simp]

theorem Matrix.replicateCol_apply {m : Type u_2} {α : Type v} {ι : Type u_6} (w : m → α) (i : m) (j : ι) : replicateCol ι w i j = w i

@[deprecated Matrix.replicateCol_apply (since := "2025-03-20")]

theorem Matrix.col_apply {m : Type u_2} {α : Type v} {ι : Type u_6} (w : m → α) (i : m) (j : ι) : replicateCol ι w i j = w i

Alias of Matrix.replicateCol_apply.

def Matrix.replicateRow {n : Type u_3} {α : Type v} (ι : Type u_6) (v : n → α) : Matrix ι n α

Matrix.replicateRow ι u is the matrix with all rows equal to the vector u.

To get a row matrix with exactly one row, Matrix.replicateRow (Fin 1) u is the canonical choice.

Equations

• Matrix.replicateRow ι v = Matrix.of fun (x : ι) (y : n) => v y

@[deprecated Matrix.replicateRow (since := "2025-03-20")]

def Matrix.row {n : Type u_3} {α : Type v} (ι : Type u_6) (v : n → α) : Matrix ι n α

Alias of Matrix.replicateRow.

---

Matrix.replicateRow ι u is the matrix with all rows equal to the vector u.

To get a row matrix with exactly one row, Matrix.replicateRow (Fin 1) u is the canonical choice.

Equations

• @Matrix.row = @Matrix.replicateRow

@[simp]

theorem Matrix.replicateRow_apply {n : Type u_3} {α : Type v} {ι : Type u_6} (v : n → α) (i : ι) (j : n) : replicateRow ι v i j = v j

@[deprecated Matrix.replicateRow_apply (since := "2025-03-20")]

theorem Matrix.row_apply {n : Type u_3} {α : Type v} {ι : Type u_6} (v : n → α) (i : ι) (j : n) : replicateRow ι v i j = v j

Alias of Matrix.replicateRow_apply.

theorem Matrix.replicateCol_injective {m : Type u_2} {α : Type v} {ι : Type u_6} [Nonempty ι] : Function.Injective (replicateCol ι)

@[deprecated Matrix.replicateCol_injective (since := "2025-03-20")]

theorem Matrix.col_injective {m : Type u_2} {α : Type v} {ι : Type u_6} [Nonempty ι] : Function.Injective (replicateCol ι)

Alias of Matrix.replicateCol_injective.

@[simp]

theorem Matrix.replicateCol_inj {m : Type u_2} {α : Type v} {ι : Type u_6} [Nonempty ι] {v w : m → α} : replicateCol ι v = replicateCol ι w ↔ v = w

@[deprecated Matrix.replicateCol_inj (since := "2025-03-20")]

theorem Matrix.col_inj {m : Type u_2} {α : Type v} {ι : Type u_6} [Nonempty ι] {v w : m → α} : replicateCol ι v = replicateCol ι w ↔ v = w

Alias of Matrix.replicateCol_inj.

@[simp]

theorem Matrix.replicateCol_zero {m : Type u_2} {α : Type v} {ι : Type u_6} [Zero α] : replicateCol ι 0 = 0

@[deprecated Matrix.replicateCol_zero (since := "2025-03-20")]

theorem Matrix.col_zero {m : Type u_2} {α : Type v} {ι : Type u_6} [Zero α] : replicateCol ι 0 = 0

Alias of Matrix.replicateCol_zero.

@[simp]

theorem Matrix.replicateCol_eq_zero {m : Type u_2} {α : Type v} {ι : Type u_6} [Zero α] [Nonempty ι] (v : m → α) : replicateCol ι v = 0 ↔ v = 0

@[deprecated Matrix.replicateCol_eq_zero (since := "2025-03-20")]

theorem Matrix.col_eq_zero {m : Type u_2} {α : Type v} {ι : Type u_6} [Zero α] [Nonempty ι] (v : m → α) : replicateCol ι v = 0 ↔ v = 0

Alias of Matrix.replicateCol_eq_zero.

@[simp]

theorem Matrix.replicateCol_add {m : Type u_2} {α : Type v} {ι : Type u_6} [Add α] (v w : m → α) : replicateCol ι (v + w) = replicateCol ι v + replicateCol ι w

@[deprecated Matrix.replicateCol_add (since := "2025-03-20")]

theorem Matrix.col_add {m : Type u_2} {α : Type v} {ι : Type u_6} [Add α] (v w : m → α) : replicateCol ι (v + w) = replicateCol ι v + replicateCol ι w

Alias of Matrix.replicateCol_add.

@[simp]

theorem Matrix.replicateCol_smul {m : Type u_2} {R : Type u_5} {α : Type v} {ι : Type u_6} [SMul R α] (x : R) (v : m → α) : replicateCol ι (x • v) = x • replicateCol ι v

@[deprecated Matrix.replicateCol_smul (since := "2025-03-20")]

theorem Matrix.col_smul {m : Type u_2} {R : Type u_5} {α : Type v} {ι : Type u_6} [SMul R α] (x : R) (v : m → α) : replicateCol ι (x • v) = x • replicateCol ι v

Alias of Matrix.replicateCol_smul.

theorem Matrix.replicateRow_injective {n : Type u_3} {α : Type v} {ι : Type u_6} [Nonempty ι] : Function.Injective (replicateRow ι)

@[deprecated Matrix.replicateRow_injective (since := "2025-03-20")]

theorem Matrix.row_injective {n : Type u_3} {α : Type v} {ι : Type u_6} [Nonempty ι] : Function.Injective (replicateRow ι)

Alias of Matrix.replicateRow_injective.

@[simp]

theorem Matrix.replicateRow_inj {n : Type u_3} {α : Type v} {ι : Type u_6} [Nonempty ι] {v w : n → α} : replicateRow ι v = replicateRow ι w ↔ v = w

@[simp]

theorem Matrix.replicateRow_zero {n : Type u_3} {α : Type v} {ι : Type u_6} [Zero α] : replicateRow ι 0 = 0

@[deprecated Matrix.replicateRow_zero (since := "2025-03-20")]

theorem Matrix.row_zero {n : Type u_3} {α : Type v} {ι : Type u_6} [Zero α] : replicateRow ι 0 = 0

Alias of Matrix.replicateRow_zero.

@[simp]

theorem Matrix.replicateRow_eq_zero {n : Type u_3} {α : Type v} {ι : Type u_6} [Zero α] [Nonempty ι] (v : n → α) : replicateRow ι v = 0 ↔ v = 0

@[deprecated Matrix.replicateRow_eq_zero (since := "2025-03-20")]

theorem Matrix.row_eq_zero {n : Type u_3} {α : Type v} {ι : Type u_6} [Zero α] [Nonempty ι] (v : n → α) : replicateRow ι v = 0 ↔ v = 0

Alias of Matrix.replicateRow_eq_zero.

@[simp]

theorem Matrix.replicateRow_add {m : Type u_2} {α : Type v} {ι : Type u_6} [Add α] (v w : m → α) : replicateRow ι (v + w) = replicateRow ι v + replicateRow ι w

@[deprecated Matrix.replicateRow_add (since := "2025-03-20")]

theorem Matrix.row_add {m : Type u_2} {α : Type v} {ι : Type u_6} [Add α] (v w : m → α) : replicateRow ι (v + w) = replicateRow ι v + replicateRow ι w

Alias of Matrix.replicateRow_add.

@[simp]

theorem Matrix.replicateRow_smul {m : Type u_2} {R : Type u_5} {α : Type v} {ι : Type u_6} [SMul R α] (x : R) (v : m → α) : replicateRow ι (x • v) = x • replicateRow ι v

@[deprecated Matrix.replicateRow_smul (since := "2025-03-20")]

theorem Matrix.row_smul {m : Type u_2} {R : Type u_5} {α : Type v} {ι : Type u_6} [SMul R α] (x : R) (v : m → α) : replicateRow ι (x • v) = x • replicateRow ι v

Alias of Matrix.replicateRow_smul.

@[simp]

theorem Matrix.transpose_replicateCol {m : Type u_2} {α : Type v} {ι : Type u_6} (v : m → α) : (replicateCol ι v).transpose = replicateRow ι v

@[deprecated Matrix.transpose_replicateCol (since := "2025-03-20")]

theorem Matrix.transpose_col {m : Type u_2} {α : Type v} {ι : Type u_6} (v : m → α) : (replicateCol ι v).transpose = replicateRow ι v

Alias of Matrix.transpose_replicateCol.

@[simp]

theorem Matrix.transpose_replicateRow {m : Type u_2} {α : Type v} {ι : Type u_6} (v : m → α) : (replicateRow ι v).transpose = replicateCol ι v

@[deprecated Matrix.transpose_replicateRow (since := "2025-03-20")]

theorem Matrix.transpose_row {m : Type u_2} {α : Type v} {ι : Type u_6} (v : m → α) : (replicateRow ι v).transpose = replicateCol ι v

Alias of Matrix.transpose_replicateRow.

@[simp]

theorem Matrix.conjTranspose_replicateCol {m : Type u_2} {α : Type v} {ι : Type u_6} [Star α] (v : m → α) : (replicateCol ι v).conjTranspose = replicateRow ι (star v)

@[deprecated Matrix.conjTranspose_replicateCol (since := "2025-03-20")]

theorem Matrix.conjTranspose_col {m : Type u_2} {α : Type v} {ι : Type u_6} [Star α] (v : m → α) : (replicateCol ι v).conjTranspose = replicateRow ι (star v)

Alias of Matrix.conjTranspose_replicateCol.

@[simp]

theorem Matrix.conjTranspose_replicateRow {m : Type u_2} {α : Type v} {ι : Type u_6} [Star α] (v : m → α) : (replicateRow ι v).conjTranspose = replicateCol ι (star v)

@[deprecated Matrix.conjTranspose_replicateRow (since := "2025-03-20")]

theorem Matrix.conjTranspose_row {m : Type u_2} {α : Type v} {ι : Type u_6} [Star α] (v : m → α) : (replicateRow ι v).conjTranspose = replicateCol ι (star v)

Alias of Matrix.conjTranspose_replicateRow.

theorem Matrix.replicateRow_vecMul {m : Type u_2} {n : Type u_3} {α : Type v} {ι : Type u_6} [Fintype m] [NonUnitalNonAssocSemiring α] (M : Matrix m n α) (v : m → α) : replicateRow ι (vecMul v M) = replicateRow ι v * M

@[deprecated Matrix.replicateRow_vecMul (since := "2025-03-20")]

theorem Matrix.row_vecMul {m : Type u_2} {n : Type u_3} {α : Type v} {ι : Type u_6} [Fintype m] [NonUnitalNonAssocSemiring α] (M : Matrix m n α) (v : m → α) : replicateRow ι (vecMul v M) = replicateRow ι v * M

Alias of Matrix.replicateRow_vecMul.

theorem Matrix.replicateCol_vecMul {m : Type u_2} {n : Type u_3} {α : Type v} {ι : Type u_6} [Fintype m] [NonUnitalNonAssocSemiring α] (M : Matrix m n α) (v : m → α) : replicateCol ι (vecMul v M) = (replicateRow ι v * M).transpose

@[deprecated Matrix.replicateCol_vecMul (since := "2025-03-20")]

theorem Matrix.col_vecMul {m : Type u_2} {n : Type u_3} {α : Type v} {ι : Type u_6} [Fintype m] [NonUnitalNonAssocSemiring α] (M : Matrix m n α) (v : m → α) : replicateCol ι (vecMul v M) = (replicateRow ι v * M).transpose

Alias of Matrix.replicateCol_vecMul.

theorem Matrix.replicateCol_mulVec {m : Type u_2} {n : Type u_3} {α : Type v} {ι : Type u_6} [Fintype n] [NonUnitalNonAssocSemiring α] (M : Matrix m n α) (v : n → α) : replicateCol ι (M.mulVec v) = M * replicateCol ι v

@[deprecated Matrix.replicateCol_mulVec (since := "2025-03-20")]

theorem Matrix.col_mulVec {m : Type u_2} {n : Type u_3} {α : Type v} {ι : Type u_6} [Fintype n] [NonUnitalNonAssocSemiring α] (M : Matrix m n α) (v : n → α) : replicateCol ι (M.mulVec v) = M * replicateCol ι v

Alias of Matrix.replicateCol_mulVec.

theorem Matrix.replicateRow_mulVec {m : Type u_2} {n : Type u_3} {α : Type v} {ι : Type u_6} [Fintype n] [NonUnitalNonAssocSemiring α] (M : Matrix m n α) (v : n → α) : replicateRow ι (M.mulVec v) = (M * replicateCol ι v).transpose

@[deprecated Matrix.replicateRow_mulVec (since := "2025-03-20")]

theorem Matrix.row_mulVec {m : Type u_2} {n : Type u_3} {α : Type v} {ι : Type u_6} [Fintype n] [NonUnitalNonAssocSemiring α] (M : Matrix m n α) (v : n → α) : replicateRow ι (M.mulVec v) = (M * replicateCol ι v).transpose

Alias of Matrix.replicateRow_mulVec.

theorem Matrix.replicateRow_mulVec_eq_const {m : Type u_2} {α : Type v} {ι : Type u_6} [Fintype m] [NonUnitalNonAssocSemiring α] (v w : m → α) : (replicateRow ι v).mulVec w = Function.const ι (v ⬝ᵥ w)

@[deprecated Matrix.replicateRow_mulVec_eq_const (since := "2025-03-20")]

theorem Matrix.row_mulVec_eq_const {m : Type u_2} {α : Type v} {ι : Type u_6} [Fintype m] [NonUnitalNonAssocSemiring α] (v w : m → α) : (replicateRow ι v).mulVec w = Function.const ι (v ⬝ᵥ w)

Alias of Matrix.replicateRow_mulVec_eq_const.

theorem Matrix.mulVec_replicateCol_eq_const {m : Type u_2} {α : Type v} {ι : Type u_6} [Fintype m] [NonUnitalNonAssocSemiring α] (v w : m → α) : vecMul v (replicateCol ι w) = Function.const ι (v ⬝ᵥ w)

@[deprecated Matrix.mulVec_replicateCol_eq_const (since := "2025-03-20")]

theorem Matrix.mulVec_col_eq_const {m : Type u_2} {α : Type v} {ι : Type u_6} [Fintype m] [NonUnitalNonAssocSemiring α] (v w : m → α) : vecMul v (replicateCol ι w) = Function.const ι (v ⬝ᵥ w)

Alias of Matrix.mulVec_replicateCol_eq_const.

theorem Matrix.replicateRow_mul_replicateCol {m : Type u_2} {α : Type v} {ι : Type u_6} [Fintype m] [Mul α] [AddCommMonoid α] (v w : m → α) : replicateRow ι v * replicateCol ι w = of fun (x x : ι) => v ⬝ᵥ w

@[deprecated Matrix.replicateRow_mul_replicateCol (since := "2025-03-20")]

theorem Matrix.row_mul_col {m : Type u_2} {α : Type v} {ι : Type u_6} [Fintype m] [Mul α] [AddCommMonoid α] (v w : m → α) : replicateRow ι v * replicateCol ι w = of fun (x x : ι) => v ⬝ᵥ w

Alias of Matrix.replicateRow_mul_replicateCol.

@[simp]

theorem Matrix.replicateRow_mul_replicateCol_apply {m : Type u_2} {α : Type v} {ι : Type u_6} [Fintype m] [Mul α] [AddCommMonoid α] (v w : m → α) (i j : ι) : (replicateRow ι v * replicateCol ι w) i j = v ⬝ᵥ w

@[deprecated Matrix.replicateRow_mul_replicateCol_apply (since := "2025-03-20")]

theorem Matrix.row_mul_col_apply {m : Type u_2} {α : Type v} {ι : Type u_6} [Fintype m] [Mul α] [AddCommMonoid α] (v w : m → α) (i j : ι) : (replicateRow ι v * replicateCol ι w) i j = v ⬝ᵥ w

Alias of Matrix.replicateRow_mul_replicateCol_apply.

@[simp]

theorem Matrix.diag_replicateCol_mul_replicateRow {n : Type u_3} {α : Type v} {ι : Type u_6} [Mul α] [AddCommMonoid α] [Unique ι] (a b : n → α) : (replicateCol ι a * replicateRow ι b).diag = a * b

@[deprecated Matrix.diag_replicateCol_mul_replicateRow (since := "2025-03-20")]

theorem Matrix.diag_col_mul_row {n : Type u_3} {α : Type v} {ι : Type u_6} [Mul α] [AddCommMonoid α] [Unique ι] (a b : n → α) : (replicateCol ι a * replicateRow ι b).diag = a * b

Alias of Matrix.diag_replicateCol_mul_replicateRow.

theorem Matrix.vecMulVec_eq {m : Type u_2} {n : Type u_3} {α : Type v} (ι : Type u_6) [Mul α] [AddCommMonoid α] [Unique ι] (w : m → α) (v : n → α) : vecMulVec w v = replicateCol ι w * replicateRow ι v

Updating rows and columns

def Matrix.updateRow {m : Type u_2} {n : Type u_3} {α : Type v} [DecidableEq m] (M : Matrix m n α) (i : m) (b : n → α) : Matrix m n α

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

Equations

• M.updateRow i b = Matrix.of (Function.update M i b)

def Matrix.updateCol {m : Type u_2} {n : Type u_3} {α : Type v} [DecidableEq n] (M : Matrix m n α) (j : n) (b : m → α) : Matrix m n α

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

Equations

• M.updateCol j b = Matrix.of fun (i : m) => Function.update (M i) j (b i)

@[deprecated Matrix.updateCol (since := "2024-12-11")]

def Matrix.updateColumn {m : Type u_2} {n : Type u_3} {α : Type v} [DecidableEq n] (M : Matrix m n α) (j : n) (b : m → α) : Matrix m n α

Alias of Matrix.updateCol.

---

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

Equations

• @Matrix.updateColumn = @Matrix.updateCol

@[simp]

theorem Matrix.updateRow_self {m : Type u_2} {n : Type u_3} {α : Type v} {M : Matrix m n α} {i : m} {b : n → α} [DecidableEq m] : M.updateRow i b i = b

@[simp]

theorem Matrix.updateCol_self {m : Type u_2} {n : Type u_3} {α : Type v} {M : Matrix m n α} {i : m} {j : n} {c : m → α} [DecidableEq n] : M.updateCol j c i j = c i

@[deprecated Matrix.updateCol_self (since := "2024-12-11")]

theorem Matrix.updateColumn_self {m : Type u_2} {n : Type u_3} {α : Type v} {M : Matrix m n α} {i : m} {j : n} {c : m → α} [DecidableEq n] : M.updateCol j c i j = c i

Alias of Matrix.updateCol_self.

@[simp]

theorem Matrix.updateRow_ne {m : Type u_2} {n : Type u_3} {α : Type v} {M : Matrix m n α} {i : m} {b : n → α} [DecidableEq m] {i' : m} (i_ne : i' ≠ i) : M.updateRow i b i' = M i'

@[simp]

theorem Matrix.updateCol_ne {m : Type u_2} {n : Type u_3} {α : Type v} {M : Matrix m n α} {i : m} {j : n} {c : m → α} [DecidableEq n] {j' : n} (j_ne : j' ≠ j) : M.updateCol j c i j' = M i j'

@[deprecated Matrix.updateCol_ne (since := "2024-12-11")]

theorem Matrix.updateColumn_ne {m : Type u_2} {n : Type u_3} {α : Type v} {M : Matrix m n α} {i : m} {j : n} {c : m → α} [DecidableEq n] {j' : n} (j_ne : j' ≠ j) : M.updateCol j c i j' = M i j'

Alias of Matrix.updateCol_ne.

theorem Matrix.updateRow_apply {m : Type u_2} {n : Type u_3} {α : Type v} {M : Matrix m n α} {i : m} {j : n} {b : n → α} [DecidableEq m] {i' : m} : M.updateRow i b i' j = if i' = i then b j else M i' j

theorem Matrix.updateCol_apply {m : Type u_2} {n : Type u_3} {α : Type v} {M : Matrix m n α} {i : m} {j : n} {c : m → α} [DecidableEq n] {j' : n} : M.updateCol j c i j' = if j' = j then c i else M i j'

@[deprecated Matrix.updateCol_apply (since := "2024-12-11")]

theorem Matrix.updateColumn_apply {m : Type u_2} {n : Type u_3} {α : Type v} {M : Matrix m n α} {i : m} {j : n} {c : m → α} [DecidableEq n] {j' : n} : M.updateCol j c i j' = if j' = j then c i else M i j'

Alias of Matrix.updateCol_apply.

@[simp]

theorem Matrix.updateCol_subsingleton {m : Type u_2} {n : Type u_3} {R : Type u_5} [Subsingleton n] (A : Matrix m n R) (i : n) (b : m → R) : A.updateCol i b = (replicateCol (Fin 1) b).submatrix id (Function.const n 0)

@[deprecated Matrix.updateCol_subsingleton (since := "2024-12-11")]

theorem Matrix.updateColumn_subsingleton {m : Type u_2} {n : Type u_3} {R : Type u_5} [Subsingleton n] (A : Matrix m n R) (i : n) (b : m → R) : A.updateCol i b = (replicateCol (Fin 1) b).submatrix id (Function.const n 0)

Alias of Matrix.updateCol_subsingleton.

@[simp]

theorem Matrix.updateRow_subsingleton {m : Type u_2} {n : Type u_3} {R : Type u_5} [Subsingleton m] (A : Matrix m n R) (i : m) (b : n → R) : A.updateRow i b = (replicateRow (Fin 1) b).submatrix (Function.const m 0) id

theorem Matrix.map_updateRow {m : Type u_2} {n : Type u_3} {α : Type v} {β : Type w} {M : Matrix m n α} {i : m} {b : n → α} [DecidableEq m] (f : α → β) : (M.updateRow i b).map f = (M.map f).updateRow i (f ∘ b)

theorem Matrix.map_updateCol {m : Type u_2} {n : Type u_3} {α : Type v} {β : Type w} {M : Matrix m n α} {j : n} {c : m → α} [DecidableEq n] (f : α → β) : (M.updateCol j c).map f = (M.map f).updateCol j (f ∘ c)

@[deprecated Matrix.map_updateCol (since := "2024-12-11")]

theorem Matrix.map_updateColumn {m : Type u_2} {n : Type u_3} {α : Type v} {β : Type w} {M : Matrix m n α} {j : n} {c : m → α} [DecidableEq n] (f : α → β) : (M.updateCol j c).map f = (M.map f).updateCol j (f ∘ c)

Alias of Matrix.map_updateCol.

theorem Matrix.updateRow_transpose {m : Type u_2} {n : Type u_3} {α : Type v} {M : Matrix m n α} {j : n} {c : m → α} [DecidableEq n] : M.transpose.updateRow j c = (M.updateCol j c).transpose

theorem Matrix.updateCol_transpose {m : Type u_2} {n : Type u_3} {α : Type v} {M : Matrix m n α} {i : m} {b : n → α} [DecidableEq m] : M.transpose.updateCol i b = (M.updateRow i b).transpose

@[deprecated Matrix.updateCol_transpose (since := "2024-12-11")]

theorem Matrix.updateColumn_transpose {m : Type u_2} {n : Type u_3} {α : Type v} {M : Matrix m n α} {i : m} {b : n → α} [DecidableEq m] : M.transpose.updateCol i b = (M.updateRow i b).transpose

Alias of Matrix.updateCol_transpose.

theorem Matrix.updateRow_conjTranspose {m : Type u_2} {n : Type u_3} {α : Type v} {M : Matrix m n α} {j : n} {c : m → α} [DecidableEq n] [Star α] : M.conjTranspose.updateRow j (star c) = (M.updateCol j c).conjTranspose

theorem Matrix.updateCol_conjTranspose {m : Type u_2} {n : Type u_3} {α : Type v} {M : Matrix m n α} {i : m} {b : n → α} [DecidableEq m] [Star α] : M.conjTranspose.updateCol i (star b) = (M.updateRow i b).conjTranspose

@[deprecated Matrix.updateCol_conjTranspose (since := "2024-12-11")]

theorem Matrix.updateColumn_conjTranspose {m : Type u_2} {n : Type u_3} {α : Type v} {M : Matrix m n α} {i : m} {b : n → α} [DecidableEq m] [Star α] : M.conjTranspose.updateCol i (star b) = (M.updateRow i b).conjTranspose

Alias of Matrix.updateCol_conjTranspose.

@[simp]

theorem Matrix.updateRow_eq_self {m : Type u_2} {n : Type u_3} {α : Type v} [DecidableEq m] (A : Matrix m n α) (i : m) : A.updateRow i (A i) = A

@[simp]

theorem Matrix.updateCol_eq_self {m : Type u_2} {n : Type u_3} {α : Type v} [DecidableEq n] (A : Matrix m n α) (i : n) : (A.updateCol i fun (j : m) => A j i) = A

@[deprecated Matrix.updateCol_eq_self (since := "2024-12-11")]

theorem Matrix.updateColumn_eq_self {m : Type u_2} {n : Type u_3} {α : Type v} [DecidableEq n] (A : Matrix m n α) (i : n) : (A.updateCol i fun (j : m) => A j i) = A

Alias of Matrix.updateCol_eq_self.

theorem Matrix.diagonal_updateCol_single {n : Type u_3} {α : Type v} [DecidableEq n] [Zero α] (v : n → α) (i : n) (x : α) : (diagonal v).updateCol i (Pi.single i x) = diagonal (Function.update v i x)

@[deprecated Matrix.diagonal_updateCol_single (since := "2024-12-11")]

theorem Matrix.diagonal_updateColumn_single {n : Type u_3} {α : Type v} [DecidableEq n] [Zero α] (v : n → α) (i : n) (x : α) : (diagonal v).updateCol i (Pi.single i x) = diagonal (Function.update v i x)

Alias of Matrix.diagonal_updateCol_single.

theorem Matrix.diagonal_updateRow_single {n : Type u_3} {α : Type v} [DecidableEq n] [Zero α] (v : n → α) (i : n) (x : α) : (diagonal v).updateRow i (Pi.single i x) = diagonal (Function.update v i x)

Updating rows and columns commutes in the obvious way with reindexing the matrix.

theorem Matrix.updateRow_submatrix_equiv {l : Type u_1} {m : Type u_2} {n : Type u_3} {o : Type u_4} {α : Type v} [DecidableEq l] [DecidableEq m] (A : Matrix m n α) (i : l) (r : o → α) (e : l ≃ m) (f : o ≃ n) : (A.submatrix ⇑e ⇑f).updateRow i r = (A.updateRow (e i) fun (j : n) => r (f.symm j)).submatrix ⇑e ⇑f

theorem Matrix.submatrix_updateRow_equiv {l : Type u_1} {m : Type u_2} {n : Type u_3} {o : Type u_4} {α : Type v} [DecidableEq l] [DecidableEq m] (A : Matrix m n α) (i : m) (r : n → α) (e : l ≃ m) (f : o ≃ n) : (A.updateRow i r).submatrix ⇑e ⇑f = (A.submatrix ⇑e ⇑f).updateRow (e.symm i) fun (i : o) => r (f i)

theorem Matrix.updateCol_submatrix_equiv {l : Type u_1} {m : Type u_2} {n : Type u_3} {o : Type u_4} {α : Type v} [DecidableEq o] [DecidableEq n] (A : Matrix m n α) (j : o) (c : l → α) (e : l ≃ m) (f : o ≃ n) : (A.submatrix ⇑e ⇑f).updateCol j c = (A.updateCol (f j) fun (i : m) => c (e.symm i)).submatrix ⇑e ⇑f

@[deprecated Matrix.updateCol_submatrix_equiv (since := "2024-12-11")]

theorem Matrix.updateColumn_submatrix_equiv {l : Type u_1} {m : Type u_2} {n : Type u_3} {o : Type u_4} {α : Type v} [DecidableEq o] [DecidableEq n] (A : Matrix m n α) (j : o) (c : l → α) (e : l ≃ m) (f : o ≃ n) : (A.submatrix ⇑e ⇑f).updateCol j c = (A.updateCol (f j) fun (i : m) => c (e.symm i)).submatrix ⇑e ⇑f

Alias of Matrix.updateCol_submatrix_equiv.

theorem Matrix.submatrix_updateCol_equiv {l : Type u_1} {m : Type u_2} {n : Type u_3} {o : Type u_4} {α : Type v} [DecidableEq o] [DecidableEq n] (A : Matrix m n α) (j : n) (c : m → α) (e : l ≃ m) (f : o ≃ n) : (A.updateCol j c).submatrix ⇑e ⇑f = (A.submatrix ⇑e ⇑f).updateCol (f.symm j) fun (i : l) => c (e i)

@[deprecated Matrix.submatrix_updateCol_equiv (since := "2024-12-11")]

theorem Matrix.submatrix_updateColumn_equiv {l : Type u_1} {m : Type u_2} {n : Type u_3} {o : Type u_4} {α : Type v} [DecidableEq o] [DecidableEq n] (A : Matrix m n α) (j : n) (c : m → α) (e : l ≃ m) (f : o ≃ n) : (A.updateCol j c).submatrix ⇑e ⇑f = (A.submatrix ⇑e ⇑f).updateCol (f.symm j) fun (i : l) => c (e i)

Alias of Matrix.submatrix_updateCol_equiv.

reindex versions of the above submatrix lemmas for convenience.

theorem Matrix.updateRow_reindex {l : Type u_1} {m : Type u_2} {n : Type u_3} {o : Type u_4} {α : Type v} [DecidableEq l] [DecidableEq m] (A : Matrix m n α) (i : l) (r : o → α) (e : m ≃ l) (f : n ≃ o) : ((reindex e f) A).updateRow i r = (reindex e f) (A.updateRow (e.symm i) fun (j : n) => r (f j))

theorem Matrix.reindex_updateRow {l : Type u_1} {m : Type u_2} {n : Type u_3} {o : Type u_4} {α : Type v} [DecidableEq l] [DecidableEq m] (A : Matrix m n α) (i : m) (r : n → α) (e : m ≃ l) (f : n ≃ o) : (reindex e f) (A.updateRow i r) = ((reindex e f) A).updateRow (e i) fun (i : o) => r (f.symm i)

theorem Matrix.updateCol_reindex {l : Type u_1} {m : Type u_2} {n : Type u_3} {o : Type u_4} {α : Type v} [DecidableEq o] [DecidableEq n] (A : Matrix m n α) (j : o) (c : l → α) (e : m ≃ l) (f : n ≃ o) : ((reindex e f) A).updateCol j c = (reindex e f) (A.updateCol (f.symm j) fun (i : m) => c (e i))

@[deprecated Matrix.updateCol_reindex (since := "2024-12-11")]

theorem Matrix.updateColumn_reindex {l : Type u_1} {m : Type u_2} {n : Type u_3} {o : Type u_4} {α : Type v} [DecidableEq o] [DecidableEq n] (A : Matrix m n α) (j : o) (c : l → α) (e : m ≃ l) (f : n ≃ o) : ((reindex e f) A).updateCol j c = (reindex e f) (A.updateCol (f.symm j) fun (i : m) => c (e i))

Alias of Matrix.updateCol_reindex.

theorem Matrix.reindex_updateCol {l : Type u_1} {m : Type u_2} {n : Type u_3} {o : Type u_4} {α : Type v} [DecidableEq o] [DecidableEq n] (A : Matrix m n α) (j : n) (c : m → α) (e : m ≃ l) (f : n ≃ o) : (reindex e f) (A.updateCol j c) = ((reindex e f) A).updateCol (f j) fun (i : l) => c (e.symm i)

@[deprecated Matrix.reindex_updateCol (since := "2024-12-11")]

theorem Matrix.reindex_updateColumn {l : Type u_1} {m : Type u_2} {n : Type u_3} {o : Type u_4} {α : Type v} [DecidableEq o] [DecidableEq n] (A : Matrix m n α) (j : n) (c : m → α) (e : m ≃ l) (f : n ≃ o) : (reindex e f) (A.updateCol j c) = ((reindex e f) A).updateCol (f j) fun (i : l) => c (e.symm i)

Alias of Matrix.reindex_updateCol.