Row and column matrices

This file provides results about row and column matrices

Main definitions

• Matrix.row r : Matrix Unit n α: a matrix with a single row
• Matrix.col c : Matrix m Unit α: a matrix with a single column
• 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.col {m : Type u_2} {α : Type v} (ι : Type u_6) (w : m → α) : Matrix m ι α

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

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

Equations

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

@[simp]

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

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

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

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

Equations

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

@[simp]

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

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

@[simp]

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

@[simp]

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

@[simp]

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

@[simp]

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

@[simp]

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) : Matrix.col ι (x • v) = x • Matrix.col ι v

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

@[simp]

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

@[simp]

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

@[simp]

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

@[simp]

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

@[simp]

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) : Matrix.row ι (x • v) = x • Matrix.row ι v

@[simp]

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

@[simp]

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

@[simp]

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

@[simp]

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

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) : Matrix.row ι (Matrix.vecMul v M) = Matrix.row ι v * M

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) : Matrix.col ι (Matrix.vecMul v M) = (Matrix.row ι v * M).transpose

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) : Matrix.col ι (M.mulVec v) = M * Matrix.col ι v

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) : Matrix.row ι (M.mulVec v) = (M * Matrix.col ι v).transpose

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

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

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

@[simp]

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

@[simp]

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

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) : Matrix.vecMulVec w v = Matrix.col ι w * Matrix.row 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.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

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

Equations

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

@[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.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.updateColumn j c i j = c i

@[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.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.updateColumn j c i j' = M i j'

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.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.updateColumn j c i j' = if j' = j then c i else M i j'

@[simp]

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.updateColumn i b = (Matrix.col (Fin 1) b).submatrix id (Function.const n 0)

@[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 = (Matrix.row (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_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.updateColumn j c).map f = (M.map f).updateColumn j (f ∘ c)

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.updateColumn j c).transpose

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.updateColumn i b = (M.updateRow i b).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.updateColumn j c).conjTranspose

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.updateColumn i (star b) = (M.updateRow i b).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.updateColumn_eq_self {m : Type u_2} {n : Type u_3} {α : Type v} [DecidableEq n] (A : Matrix m n α) (i : n) : (A.updateColumn i fun (j : m) => A j i) = A

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

theorem Matrix.diagonal_updateRow_single {n : Type u_3} {α : Type v} [DecidableEq n] [Zero α] (v : n → α) (i : n) (x : α) : (Matrix.diagonal v).updateRow i (Pi.single i x) = Matrix.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.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).updateColumn j c = (A.updateColumn (f j) fun (i : m) => c (e.symm i)).submatrix ⇑e ⇑f

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.updateColumn j c).submatrix ⇑e ⇑f = (A.submatrix ⇑e ⇑f).updateColumn (f.symm j) fun (i : l) => c (e i)

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) : ((Matrix.reindex e f) A).updateRow i r = (Matrix.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) : (Matrix.reindex e f) (A.updateRow i r) = ((Matrix.reindex e f) A).updateRow (e i) fun (i : o) => r (f.symm i)

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) : ((Matrix.reindex e f) A).updateColumn j c = (Matrix.reindex e f) (A.updateColumn (f.symm j) fun (i : m) => c (e i))

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) : (Matrix.reindex e f) (A.updateColumn j c) = ((Matrix.reindex e f) A).updateColumn (f j) fun (i : l) => c (e.symm i)