Theory Complex_Matrix

section ‹Complex matrices›

theory Complex_Matrix
  imports 
    "Jordan_Normal_Form.Matrix" 
    "Jordan_Normal_Form.Conjugate" 
    "Jordan_Normal_Form.Jordan_Normal_Form_Existence"
begin

subsection ‹Trace of a matrix›

definition trace :: "'a::ring mat ⇒ 'a" where
  "trace A = (∑ i ∈ {0 ..< dim_row A}. A $$ (i,i))"

lemma trace_zero [simp]:
  "trace (0⇩m n n) = 0"
  by (simp add: trace_def)

lemma trace_id [simp]:
  "trace (1⇩m n) = n"
  by (simp add: trace_def)

lemma trace_comm:
  fixes A B :: "'a::comm_ring mat"
  assumes A: "A ∈ carrier_mat n n" and B: "B ∈ carrier_mat n n"
  shows "trace (A * B) = trace (B * A)"
proof (simp add: trace_def)
  have "(∑i = 0..<n. (A * B) $$ (i, i)) = (∑i = 0..<n. ∑j = 0..<n. A $$ (i,j) * B $$ (j,i))"
    apply (rule sum.cong) using assms by (auto simp add: scalar_prod_def)
  also have "… = (∑j = 0..<n. ∑i = 0..<n. A $$ (i,j) * B $$ (j,i))"
    by (rule sum.swap)
  also have "… = (∑j = 0..<n. col A j ∙ row B j)"
    by (metis (no_types, lifting) A B atLeastLessThan_iff carrier_matD index_col index_row scalar_prod_def sum.cong)
  also have "… = (∑j = 0..<n. row B j ∙ col A j)"
    apply (rule sum.cong) apply auto
    apply (subst comm_scalar_prod[where n=n]) apply auto
    using assms by auto
  also have "… = (∑j = 0..<n. (B * A) $$ (j, j))"
    apply (rule sum.cong) using assms by auto
  finally show "(∑i = 0..<dim_row A. (A * B) $$ (i, i)) = (∑i = 0..<dim_row B. (B * A) $$ (i, i))"
    using A B by auto
qed

lemma trace_add_linear:
  fixes A B :: "'a::comm_ring mat"
  assumes A: "A ∈ carrier_mat n n" and B: "B ∈ carrier_mat n n"
  shows "trace (A + B) = trace A + trace B" (is "?lhs = ?rhs")
proof -
  have "?lhs = (∑i=0..<n. A$$(i, i) + B$$(i, i))" unfolding trace_def using A B by auto
  also have "… = (∑i=0..<n. A$$(i, i)) + (∑i=0..<n. B$$(i, i))" by (auto simp add: sum.distrib)
  finally have l: "?lhs = (∑i=0..<n. A$$(i, i)) + (∑i=0..<n. B$$(i, i))".
  have r: "?rhs = (∑i=0..<n. A$$(i, i)) + (∑i=0..<n. B$$(i, i))" unfolding trace_def using A B by auto
  from l r show ?thesis by auto
qed

lemma trace_minus_linear:
  fixes A B :: "'a::comm_ring mat"
  assumes A: "A ∈ carrier_mat n n" and B: "B ∈ carrier_mat n n"
  shows "trace (A - B) = trace A - trace B" (is "?lhs = ?rhs")
proof -
  have "?lhs = (∑i=0..<n. A$$(i, i) - B$$(i, i))" unfolding trace_def using A B by auto
  also have "… = (∑i=0..<n. A$$(i, i)) - (∑i=0..<n. B$$(i, i))" by (auto simp add: sum_subtractf)
  finally have l: "?lhs = (∑i=0..<n. A$$(i, i)) - (∑i=0..<n. B$$(i, i))".
  have r: "?rhs = (∑i=0..<n. A$$(i, i)) - (∑i=0..<n. B$$(i, i))" unfolding trace_def using A B by auto
  from l r show ?thesis by auto
qed

lemma trace_smult: 
  assumes "A ∈ carrier_mat n n"
  shows "trace (c ⋅⇩m A) = c * trace A"
proof -
  have "trace (c ⋅⇩m A) = (∑i = 0..<dim_row A. c * A $$ (i, i))" unfolding trace_def using assms by auto
  also have "… = c * (∑i = 0..<dim_row A. A $$ (i, i))"
    by (simp add: sum_distrib_left)
  also have "… = c * trace A" unfolding trace_def by auto
  ultimately show ?thesis by auto
qed

subsection ‹Conjugate of a vector›

lemma conjugate_scalar_prod:
  fixes v w :: "'a::conjugatable_ring vec"
  assumes "dim_vec v = dim_vec w"
  shows "conjugate (v ∙ w) = conjugate v ∙ conjugate w"
  using assms by (simp add: scalar_prod_def sum_conjugate conjugate_dist_mul)

subsection ‹Inner product›

abbreviation inner_prod :: "'a vec ⇒ 'a vec ⇒ 'a :: conjugatable_ring"
  where "inner_prod v w ≡ w ∙c v"

lemma conjugate_scalar_prod_Im [simp]:
  "Im (v ∙c v) = 0"
  by (simp add: scalar_prod_def conjugate_vec_def sum.neutral)

lemma conjugate_scalar_prod_Re [simp]:
  "Re (v ∙c v) ≥ 0"
  by (simp add: scalar_prod_def conjugate_vec_def sum_nonneg)

lemma self_cscalar_prod_geq_0:
  fixes v ::  "'a::conjugatable_ordered_field vec"
  shows "v ∙c v ≥ 0"
  by (auto simp add: scalar_prod_def, rule sum_nonneg, rule conjugate_square_positive)

lemma inner_prod_distrib_left:
  fixes u v w :: "('a::conjugatable_field) vec"
  assumes dimu: "u ∈ carrier_vec n" and dimv:"v ∈ carrier_vec n" and dimw: "w ∈ carrier_vec n" 
  shows "inner_prod (v + w) u = inner_prod v u + inner_prod w u" (is "?lhs = ?rhs")
proof -
  have dimcv: "conjugate v ∈ carrier_vec n" and dimcw: "conjugate w ∈ carrier_vec n" using assms by auto
  have dimvw: "conjugate (v + w) ∈ carrier_vec n" using assms by auto
  have "u ∙ (conjugate (v + w)) = u ∙ conjugate v + u ∙ conjugate w"
    using dimv dimw dimu dimcv dimcw 
    by (metis conjugate_add_vec scalar_prod_add_distrib)
  then show ?thesis by auto
qed

lemma inner_prod_distrib_right:
  fixes u v w :: "('a::conjugatable_field) vec"
  assumes dimu: "u ∈ carrier_vec n" and dimv:"v ∈ carrier_vec n" and dimw: "w ∈ carrier_vec n" 
  shows "inner_prod u (v + w) = inner_prod u v + inner_prod u w" (is "?lhs = ?rhs")
proof -
  have dimvw: "v + w ∈ carrier_vec n" using assms by auto
  have dimcu: "conjugate u ∈ carrier_vec n" using assms by auto
  have "(v + w) ∙ (conjugate u) = v ∙ conjugate u + w ∙ conjugate u"
    apply (simp add: comm_scalar_prod[OF dimvw dimcu])
    apply (simp add: scalar_prod_add_distrib[OF dimcu dimv dimw])
    apply (insert dimv dimw dimcu, simp add: comm_scalar_prod[of _ n])
    done
  then show ?thesis by auto
qed                

lemma inner_prod_minus_distrib_right:
  fixes u v w :: "('a::conjugatable_field) vec"
  assumes dimu: "u ∈ carrier_vec n" and dimv:"v ∈ carrier_vec n" and dimw: "w ∈ carrier_vec n" 
  shows "inner_prod u (v - w) = inner_prod u v - inner_prod u w" (is "?lhs = ?rhs")
proof -
  have dimvw: "v - w ∈ carrier_vec n" using assms by auto
  have dimcu: "conjugate u ∈ carrier_vec n" using assms by auto
  have "(v - w) ∙ (conjugate u) = v ∙ conjugate u - w ∙ conjugate u"
    apply (simp add: comm_scalar_prod[OF dimvw dimcu])
    apply (simp add: scalar_prod_minus_distrib[OF dimcu dimv dimw])
    apply (insert dimv dimw dimcu, simp add: comm_scalar_prod[of _ n])
    done
  then show ?thesis by auto
qed                

lemma inner_prod_smult_right:
  fixes u v :: "complex vec"
  assumes dimu: "u ∈ carrier_vec n" and dimv:"v ∈ carrier_vec n" 
  shows "inner_prod (a ⋅⇩v u) v = conjugate a * inner_prod u v" (is "?lhs = ?rhs")
  using assms apply (simp add: scalar_prod_def conjugate_dist_mul) 
  apply (subst sum_distrib_left) by (rule sum.cong, auto)

lemma inner_prod_smult_left:
  fixes u v :: "complex vec"
  assumes dimu: "u ∈ carrier_vec n" and dimv: "v ∈ carrier_vec n" 
  shows "inner_prod u (a ⋅⇩v v) = a * inner_prod u v" (is "?lhs = ?rhs")
  using assms apply (simp add: scalar_prod_def) 
  apply (subst sum_distrib_left) by (rule sum.cong, auto)

lemma inner_prod_smult_left_right:
  fixes u v :: "complex vec"
  assumes dimu: "u ∈ carrier_vec n" and dimv: "v ∈ carrier_vec n" 
  shows "inner_prod (a ⋅⇩v u) (b ⋅⇩v v) = conjugate a * b  * inner_prod u v" (is "?lhs = ?rhs")
  using assms apply (simp add: scalar_prod_def) 
  apply (subst sum_distrib_left) by (rule sum.cong, auto)

lemma inner_prod_swap:
  fixes x y :: "complex vec"
  assumes "y ∈ carrier_vec n" and "x ∈ carrier_vec n" 
  shows "inner_prod y x = conjugate (inner_prod x y)"
  apply (simp add: scalar_prod_def)
  apply (rule sum.cong) using assms by auto

text ‹Cauchy-Schwarz theorem for complex vectors. This is analogous to aux\_Cauchy
  and Cauchy\_Schwarz\_ineq in Generalizations2.thy in QR\_Decomposition. Consider
  merging and moving to Isabelle library.›
lemma aux_Cauchy:
  fixes x y :: "complex vec"
  assumes "x ∈ carrier_vec n" and "y ∈ carrier_vec n"
  shows "0 ≤ inner_prod x x + a * (inner_prod x y) + (cnj a) * ((cnj (inner_prod x y)) + a * (inner_prod y y))"
proof -
  have "(inner_prod (x+ a ⋅⇩v y) (x+a ⋅⇩v y)) = (inner_prod (x+a ⋅⇩v y) x) + (inner_prod (x+a ⋅⇩v y) (a ⋅⇩v y))" 
    apply (subst inner_prod_distrib_right) using assms by auto
  also have "… = inner_prod x x + (a) * (inner_prod x y) + cnj a * ((cnj (inner_prod x y)) + (a) * (inner_prod y y))"
    apply (subst (1 2) inner_prod_distrib_left[of _ n]) apply (auto simp add: assms)
    apply (subst (1 2) inner_prod_smult_right[of _ n]) apply (auto simp add: assms)
    apply (subst inner_prod_smult_left[of _ n]) apply (auto simp add: assms)
    apply (subst inner_prod_swap[of y n x]) apply (auto simp add: assms)
    unfolding distrib_left
    by auto
  finally show ?thesis by (metis self_cscalar_prod_geq_0)
qed

lemma Cauchy_Schwarz_complex_vec:
  fixes x y :: "complex vec"
  assumes "x ∈ carrier_vec n" and "y ∈ carrier_vec n"
  shows "inner_prod x y * inner_prod y x ≤ inner_prod x x * inner_prod y y"
proof -
  define cnj_a where "cnj_a = - (inner_prod x y)/ cnj (inner_prod y y)"
  define a where "a = cnj (cnj_a)"
  have cnj_rw: "(cnj a) = cnj_a" 
    unfolding a_def by (simp)
  have rw_0: "cnj (inner_prod x y) + a * (inner_prod y y) = 0"
    unfolding a_def cnj_a_def using assms(1) assms(2) conjugate_square_eq_0_vec by fastforce
  have "0 ≤  (inner_prod x x + a * (inner_prod x y) + (cnj a) * ((cnj (inner_prod x y)) + a * (inner_prod y y)))"
    using aux_Cauchy assms by auto
  also have "… =  (inner_prod x x + a * (inner_prod x y))" unfolding rw_0 by auto
  also have "… =  (inner_prod x x - (inner_prod x y) * cnj (inner_prod x y) / (inner_prod y y))" 
  unfolding a_def cnj_a_def by simp
  finally have " 0 ≤  (inner_prod x x - (inner_prod x y) * cnj (inner_prod x y) / (inner_prod y y)) " .
  hence "0 ≤ (inner_prod x x - (inner_prod x y) * cnj (inner_prod x y) / (inner_prod y y)) * (inner_prod y y)"
    by (auto simp: less_eq_complex_def)
  also have "… = ((inner_prod x x)*(inner_prod y y) - (inner_prod x y) * cnj (inner_prod x y))"
    by (smt add.inverse_neutral add_diff_cancel diff_0 diff_divide_eq_iff divide_cancel_right mult_eq_0_iff nonzero_mult_div_cancel_right rw_0)
  finally have "(inner_prod x y) * cnj (inner_prod x y) ≤ (inner_prod x x)*(inner_prod y y)" by auto
  then show ?thesis 
    apply (subst inner_prod_swap[of y n x]) by (auto simp add: assms)
qed

subsection ‹Hermitian adjoint of a matrix›

abbreviation adjoint where "adjoint ≡ mat_adjoint"

lemma adjoint_dim_row [simp]:
  "dim_row (adjoint A) = dim_col A" by (simp add: mat_adjoint_def)

lemma adjoint_dim_col [simp]:
  "dim_col (adjoint A) = dim_row A" by (simp add: mat_adjoint_def)

lemma adjoint_dim:
  "A ∈ carrier_mat n n ⟹ adjoint A ∈ carrier_mat n n"
  using adjoint_dim_col adjoint_dim_row by blast

lemma adjoint_def:
  "adjoint A = mat (dim_col A) (dim_row A) (λ(i,j). conjugate (A $$ (j,i)))"
  unfolding mat_adjoint_def mat_of_rows_def by auto

lemma adjoint_eval:
  assumes "i < dim_col A" "j < dim_row A"
  shows "(adjoint A) $$ (i,j) = conjugate (A $$ (j,i))"
  using assms by (simp add: adjoint_def)

lemma adjoint_row:
  assumes "i < dim_col A"
  shows "row (adjoint A) i = conjugate (col A i)"
  apply (rule eq_vecI)
  using assms by (auto simp add: adjoint_eval)

lemma adjoint_col:
  assumes "i < dim_row A"
  shows "col (adjoint A) i = conjugate (row A i)"
  apply (rule eq_vecI)
  using assms by (auto simp add: adjoint_eval)

text ‹The identity <v, A w> = <A* v, w>›
lemma adjoint_def_alter:
  fixes v w :: "'a::conjugatable_field vec"
    and A :: "'a::conjugatable_field mat"
  assumes dims: "v ∈ carrier_vec n" "w ∈ carrier_vec m" "A ∈ carrier_mat n m"
  shows "inner_prod v (A *⇩v w) = inner_prod (adjoint A *⇩v v) w" (is "?lhs = ?rhs")
proof -
  from dims have "?lhs = (∑i=0..<dim_vec v. (∑j=0..<dim_vec w.
                conjugate (v$i) * A$$(i, j) * w$j))"
    apply (simp add: scalar_prod_def sum_distrib_right )
    apply (rule sum.cong, simp)
    apply (rule sum.cong, auto)
    done
  moreover from assms have "?rhs = (∑i=0..<dim_vec v. (∑j=0..<dim_vec w.
                conjugate (v$i) * A$$(i, j) * w$j))"
    apply (simp add: scalar_prod_def  adjoint_eval 
                     sum_conjugate conjugate_dist_mul sum_distrib_left)
    apply (subst sum.swap[where ?A = "{0..<n}"])
    apply (rule sum.cong, simp)
    apply (rule sum.cong, auto)
    done
  ultimately show ?thesis by simp
qed

lemma adjoint_one:
  shows "adjoint (1⇩m n) = (1⇩m n::complex mat)"
  apply (rule eq_matI) 
  by (auto simp add: adjoint_eval)

lemma adjoint_scale:
  fixes A :: "'a::conjugatable_field mat"
  shows "adjoint (a ⋅⇩m A) = (conjugate a) ⋅⇩m adjoint A"
  apply (rule eq_matI) using conjugatable_ring_class.conjugate_dist_mul
  by (auto simp add: adjoint_eval)

lemma adjoint_add:
  fixes A B :: "'a::conjugatable_field mat"
  assumes "A ∈ carrier_mat n m" "B ∈ carrier_mat n m"
  shows "adjoint (A + B) = adjoint A + adjoint B"
  apply (rule eq_matI)
  using assms conjugatable_ring_class.conjugate_dist_add 
  by( auto simp add: adjoint_eval)

lemma adjoint_minus:
  fixes A B :: "'a::conjugatable_field mat"
  assumes "A ∈ carrier_mat n m" "B ∈ carrier_mat n m"
  shows "adjoint (A - B) = adjoint A - adjoint B"
  apply (rule eq_matI)
  using assms apply(auto simp add: adjoint_eval)
  by (metis add_uminus_conv_diff conjugate_dist_add conjugate_neg)

lemma adjoint_mult:
  fixes A B :: "'a::conjugatable_field mat"
  assumes "A ∈ carrier_mat n m" "B ∈ carrier_mat m l"
  shows "adjoint (A * B) = adjoint B * adjoint A"
proof (rule eq_matI, auto simp add: adjoint_eval adjoint_row adjoint_col)
  fix i j
  assume "i < dim_col B" "j < dim_row A"
  show "conjugate (row A j ∙ col B i) = conjugate (col B i) ∙ conjugate (row A j)"
    using assms apply (simp add: conjugate_scalar_prod)
    apply (subst comm_scalar_prod[where n="dim_row B"])
    by (auto simp add: carrier_vecI)
qed

lemma adjoint_adjoint:
  fixes A :: "'a::conjugatable_field mat"
  shows "adjoint (adjoint A) = A"
  by (rule eq_matI, auto simp add: adjoint_eval)

lemma trace_adjoint_positive:
  fixes A :: "complex mat"
  shows "trace (A * adjoint A) ≥ 0"
  apply (auto simp add: trace_def adjoint_col)
  apply (rule sum_nonneg) by auto

subsection ‹Algebraic manipulations on matrices›

lemma right_add_zero_mat[simp]:
  "(A :: 'a :: monoid_add mat) ∈ carrier_mat nr nc ⟹ A + 0⇩m nr nc = A"
  by (intro eq_matI, auto)

lemma add_carrier_mat':
  "A ∈ carrier_mat nr nc ⟹ B ∈ carrier_mat nr nc ⟹ A + B ∈ carrier_mat nr nc"
  by simp

lemma minus_carrier_mat':
  "A ∈ carrier_mat nr nc ⟹ B ∈ carrier_mat nr nc ⟹ A - B ∈ carrier_mat nr nc"
  by auto

lemma swap_plus_mat:
  fixes A B C :: "'a::semiring_1 mat"
  assumes "A ∈ carrier_mat n n" "B ∈ carrier_mat n n" "C ∈ carrier_mat n n"
  shows "A + B + C = A + C + B"
  by (metis assms assoc_add_mat comm_add_mat)

lemma uminus_mat:
  fixes A :: "complex mat"
  assumes "A ∈ carrier_mat n n"
  shows "-A = (-1) ⋅⇩m A"
  by auto

ML_file "mat_alg.ML"
method_setup mat_assoc = ‹mat_assoc_method›
  "Normalization of expressions on matrices"

lemma mat_assoc_test:
  fixes A B C D :: "complex mat"
  assumes "A ∈ carrier_mat n n" "B ∈ carrier_mat n n" "C ∈ carrier_mat n n" "D ∈ carrier_mat n n"
  shows
    "(A * B) * (C * D) = A * B * C * D"
    "adjoint (A * adjoint B) * C = B * (adjoint A * C)"
    "A * 1⇩m n * 1⇩m n * B * 1⇩m n = A * B"
    "(A - B) + (B - C) = A + (-B) + B + (-C)"
    "A + (B - C) = A + B - C"
    "A - (B + C + D) = A - B - C - D"
    "(A + B) * (B + C) = A * B + B * B + A * C + B * C"
    "A - B = A + (-1) ⋅⇩m B"
    "A * (B - C) * D = A * B * D - A * C * D"
    "trace (A * B * C) = trace (B * C * A)"
    "trace (A * B * C * D) = trace (C * D * A * B)"
    "trace (A + B * C) = trace A + trace (C * B)"
    "A + B = B + A"
    "A + B + C = C + B + A"
    "A + B + (C + D) = A + C + (B + D)"
  using assms by (mat_assoc n)+

subsection ‹Hermitian matrices›

text ‹A Hermitian matrix is a matrix that is equal to its Hermitian adjoint.›
definition hermitian :: "'a::conjugatable_field mat ⇒ bool" where
  "hermitian A ⟷ (adjoint A = A)"

lemma hermitian_one:
  shows "hermitian ((1⇩m n)::('a::conjugatable_field mat))"
  unfolding hermitian_def 
proof-
  have "conjugate (1::'a) = 1"   
    apply (subst mult_1_right[symmetric, of "conjugate 1"])
    apply (subst conjugate_id[symmetric, of "conjugate 1 * 1"])
    apply (subst conjugate_dist_mul)
    apply auto
    done
  then show "adjoint ((1⇩m n)::('a::conjugatable_field mat)) = (1⇩m n)" 
    by (auto simp add: adjoint_eval) 
qed

subsection ‹Inverse matrices›

lemma inverts_mat_symm:
  fixes A B :: "'a::field mat"
  assumes dim: "A ∈ carrier_mat n n" "B ∈ carrier_mat n n"
    and AB: "inverts_mat A B"
  shows "inverts_mat B A"
proof -
  have "A * B = 1⇩m n" using dim AB unfolding inverts_mat_def by auto
  with dim have "B * A = 1⇩m n"  by (rule mat_mult_left_right_inverse)
  then show "inverts_mat B A" using dim inverts_mat_def by auto
qed

lemma inverts_mat_unique:
  fixes A B C :: "'a::field mat"
  assumes dim: "A ∈ carrier_mat n n" "B ∈ carrier_mat n n" "C ∈ carrier_mat n n" 
    and AB: "inverts_mat A B" and AC: "inverts_mat A C"
  shows "B = C"
proof -
  have AB1: "A * B = 1⇩m n" using AB dim unfolding inverts_mat_def by auto
  have "A * C = 1⇩m n" using AC dim unfolding inverts_mat_def by auto
  then have CA1: "C * A = 1⇩m n" using mat_mult_left_right_inverse[of A n C] dim by auto
  then have "C = C * 1⇩m n" using dim by auto
  also have "… = C * (A * B)" using AB1 by auto
  also have "… = (C * A) * B" using dim by auto
  also have "… = 1⇩m n * B" using CA1 by auto
  also have "… = B" using dim by auto
  finally show "B = C" ..
qed

subsection ‹Unitary matrices›

text ‹A unitary matrix is a matrix whose Hermitian adjoint is also its inverse.›
definition unitary :: "'a::conjugatable_field mat ⇒ bool" where
  "unitary A ⟷ A ∈ carrier_mat (dim_row A) (dim_row A) ∧ inverts_mat A (adjoint A)"

lemma unitaryD2:
  assumes "A ∈ carrier_mat n n"
  shows "unitary A ⟹ inverts_mat (adjoint A) A"
  using assms adjoint_dim inverts_mat_symm unitary_def by blast

lemma unitary_simps [simp]:
  "A ∈ carrier_mat n n ⟹ unitary A ⟹ adjoint A * A = 1⇩m n"
  "A ∈ carrier_mat n n ⟹ unitary A ⟹ A * adjoint A = 1⇩m n"
  apply (metis adjoint_dim_row carrier_matD(2) inverts_mat_def unitaryD2)
  by (simp add: inverts_mat_def unitary_def)

lemma unitary_adjoint [simp]:
  assumes "A ∈ carrier_mat n n" "unitary A"
  shows "unitary (adjoint A)" 
  unfolding unitary_def
  using  adjoint_dim[OF assms(1)] assms by (auto simp add: unitaryD2[OF assms] adjoint_adjoint)

lemma unitary_one:
  shows "unitary ((1⇩m n)::('a::conjugatable_field mat))"
  unfolding unitary_def 
proof -
  define I where I_def[simp]: "I ≡ ((1⇩m n)::('a::conjugatable_field mat))"
  have dim: "I ∈ carrier_mat n n" by auto
  have "hermitian I" using hermitian_one  by auto
  hence "adjoint I = I" using hermitian_def by auto
  with dim show "I ∈ carrier_mat (dim_row I) (dim_row I) ∧ inverts_mat I (adjoint I)" 
    unfolding inverts_mat_def using dim by auto
qed

lemma unitary_zero:
  fixes A :: "'a::conjugatable_field mat"
  assumes "A ∈ carrier_mat 0 0"
  shows "unitary A"
  unfolding unitary_def inverts_mat_def Let_def using assms by auto

lemma unitary_elim:
  assumes dims: "A ∈ carrier_mat n n" "B ∈ carrier_mat n n" "P ∈ carrier_mat n n"
    and uP: "unitary P" and eq: "P * A * adjoint P = P * B * adjoint P"
  shows "A = B"
proof -
  have dimaP: "adjoint P ∈ carrier_mat n n" using dims by auto
  have iv: "inverts_mat P (adjoint P)" using uP unitary_def by auto
  then have "P * (adjoint P) = 1⇩m n" using inverts_mat_def dims by auto
  then have aPP: "adjoint P * P = 1⇩m n" using mat_mult_left_right_inverse[OF dims(3) dimaP] by auto
  have "adjoint P * (P * A * adjoint P) * P = (adjoint P * P) * A * (adjoint P * P)" 
    using dims dimaP by (mat_assoc n)
  also have "… = 1⇩m n * A * 1⇩m n" using aPP by auto
  also have "… = A" using dims by auto
  finally have eqA: "A = adjoint P * (P * A * adjoint P) * P" ..
  have "adjoint P * (P * B * adjoint P) * P = (adjoint P * P) * B * (adjoint P * P)" 
    using dims dimaP by (mat_assoc n)
  also have "… = 1⇩m n * B * 1⇩m n" using aPP by auto
  also have "… = B" using dims by auto
  finally have eqB: "B = adjoint P * (P * B * adjoint P) * P" ..
  then show ?thesis using eqA eqB eq by auto
qed

lemma unitary_is_corthogonal:
  fixes U :: "'a::conjugatable_field mat"
  assumes dim: "U ∈ carrier_mat n n" 
    and U: "unitary U"
  shows "corthogonal_mat U"
  unfolding corthogonal_mat_def Let_def
proof (rule conjI)
  have dima: "adjoint U ∈ carrier_mat n n" using dim by auto
  have aUU: "mat_adjoint U * U = (1⇩m n)"
    apply (insert U[unfolded unitary_def] dim dima, drule conjunct2)
    apply (drule inverts_mat_symm[of "U", OF dim dima], unfold inverts_mat_def, auto)
    done
  then show "diagonal_mat (mat_adjoint U * U)"
    by (simp add: diagonal_mat_def)
  show "∀i<dim_col U. (mat_adjoint U * U) $$ (i, i) ≠ 0" using dim by (simp add: aUU)
qed

lemma unitary_times_unitary:
  fixes P Q :: "'a:: conjugatable_field mat"
  assumes dim: "P ∈ carrier_mat n n" "Q ∈ carrier_mat n n"
    and uP: "unitary P" and uQ: "unitary Q"
  shows "unitary (P * Q)"
proof -
  have dim_pq: "P * Q ∈ carrier_mat n n" using dim by auto
  have "(P * Q) * adjoint (P * Q) = P * (Q * adjoint Q) * adjoint P" using dim by (mat_assoc n)
  also have "… = P * (1⇩m n) * adjoint P" using uQ dim by auto
  also have "… = P * adjoint P" using dim by (mat_assoc n)
  also have "… = 1⇩m n" using uP dim by simp
  finally have "(P * Q) * adjoint (P * Q) = 1⇩m n" by auto
  hence "inverts_mat (P * Q) (adjoint (P * Q))" 
    using inverts_mat_def dim_pq by auto
  thus "unitary (P*Q)" using unitary_def dim_pq by auto
qed

lemma unitary_operator_keep_trace:
  fixes U A :: "complex mat"
  assumes dU: "U ∈ carrier_mat n n" and dA: "A ∈ carrier_mat n n" and u: "unitary U"
  shows "trace A = trace (adjoint U * A * U)"
proof -
  have u': "U * adjoint U = 1⇩m n" using u unfolding unitary_def inverts_mat_def using dU by auto
  have "trace (adjoint U * A * U) = trace (U * adjoint U * A)" using dU dA by (mat_assoc n)
  also have "… = trace A" using u' dA by auto
  finally show ?thesis by auto
qed

subsection ‹Normalization of vectors›

definition vec_norm :: "complex vec ⇒ complex" where
  "vec_norm v ≡ csqrt (v ∙c v)"

lemma vec_norm_geq_0:
  fixes v :: "complex vec"
  shows "vec_norm v ≥ 0"
  unfolding vec_norm_def by (insert self_cscalar_prod_geq_0[of v], simp add: less_eq_complex_def)

lemma vec_norm_zero:
  fixes v ::  "complex vec"
  assumes dim: "v ∈ carrier_vec n"
  shows "vec_norm v = 0 ⟷ v = 0⇩v n"
  unfolding vec_norm_def
  by (subst conjugate_square_eq_0_vec[OF dim, symmetric], rule csqrt_eq_0)

lemma vec_norm_ge_0:
  fixes v ::  "complex vec"
  assumes dim_v: "v ∈ carrier_vec n" and neq0: "v ≠ 0⇩v n"
  shows "vec_norm v > 0"
proof -
  have geq: "vec_norm v ≥ 0" using vec_norm_geq_0 by auto
  have neq: "vec_norm v ≠ 0" 
    apply (insert dim_v neq0)
    apply (drule vec_norm_zero, auto)
    done
  show ?thesis using neq geq by (rule dual_order.not_eq_order_implies_strict)
qed

definition vec_normalize :: "complex vec ⇒ complex vec" where
  "vec_normalize v = (if (v = 0⇩v (dim_vec v)) then v else 1 / (vec_norm v) ⋅⇩v v)"

lemma normalized_vec_dim[simp]:
  assumes "(v::complex vec) ∈ carrier_vec n"
  shows "vec_normalize v ∈ carrier_vec n"
  unfolding vec_normalize_def using assms by auto

lemma vec_eq_norm_smult_normalized:
  shows "v = vec_norm v ⋅⇩v vec_normalize v"
proof (cases "v = 0⇩v (dim_vec v)")
  define n where "n = dim_vec v"
  then have dimv: "v ∈ carrier_vec n" by auto
  then have dimnv: "vec_normalize v ∈ carrier_vec n" by auto
  {
    case True
    then have v0: "v = 0⇩v n" using n_def by auto
    then have n0: "vec_norm v = 0" using vec_norm_def by auto
    have "vec_norm v ⋅⇩v vec_normalize v = 0⇩v n" 
      unfolding smult_vec_def by (auto simp add: n0 carrier_vecD[OF dimnv])
    then show ?thesis using v0 by auto
    next
    case False
    then have v: "v ≠ 0⇩v n" using n_def by auto
    then have ge0: "vec_norm v > 0" using vec_norm_ge_0 dimv by auto
    have "vec_normalize v = (1 / vec_norm v) ⋅⇩v v" using False vec_normalize_def by auto
    then have "vec_norm v ⋅⇩v vec_normalize v = (vec_norm v * (1 / vec_norm v)) ⋅⇩v v"
      using smult_smult_assoc by auto
    also have "… = v" using ge0 by auto
    finally have "v = vec_norm v ⋅⇩v vec_normalize v"..
    then show "v = vec_norm v ⋅⇩v vec_normalize v" using v by auto
  }
qed

lemma normalized_cscalar_prod:
  fixes v w :: "complex vec"
  assumes dim_v: "v ∈ carrier_vec n" and dim_w: "w ∈ carrier_vec n"
  shows "v ∙c w = (vec_norm v * vec_norm w) * (vec_normalize v ∙c vec_normalize w)"
  unfolding vec_normalize_def apply (split if_split, split if_split)
proof (intro conjI impI)
  note dim0 = dim_v dim_w
  have dim: "dim_vec v = n" "dim_vec w = n" using dim0 by auto
  {
    assume "w = 0⇩v n" "v = 0⇩v n"
    then have lhs: "v ∙c w = 0" by auto
    then moreover have rhs: "vec_norm v * vec_norm w * (v ∙c w) = 0" by auto
    ultimately have "v ∙c w = vec_norm v * vec_norm w * (v ∙c w)" by auto
  }
  with dim show "w = 0⇩v (dim_vec w) ⟹ v = 0⇩v (dim_vec v) ⟹ v ∙c w = vec_norm v * vec_norm w * (v ∙c w)" by auto
  {
    assume asm: "w = 0⇩v n" "v ≠ 0⇩v n"
    then have w0: "conjugate w = 0⇩v n" by auto
    with dim0 have "(1 / vec_norm v ⋅⇩v v) ∙c w = 0" by auto
    then moreover have rhs: "vec_norm v * vec_norm w * ((1 / vec_norm v ⋅⇩v v) ∙c w) = 0" by auto
    moreover have "v ∙c w = 0" using w0 dim0 by auto
    ultimately have "v ∙c w = vec_norm v * vec_norm w * ((1 / vec_norm v ⋅⇩v v) ∙c w)" by auto
  }
  with dim show "w = 0⇩v (dim_vec w) ⟹ v ≠ 0⇩v (dim_vec v) ⟹ v ∙c w = vec_norm v * vec_norm w * ((1 / vec_norm v ⋅⇩v v) ∙c w)" by auto
  {
    assume asm: "w ≠ 0⇩v n" "v = 0⇩v n"
    with dim0 have "v ∙c (1 / vec_norm w ⋅⇩v w) = 0" by auto
    then moreover have rhs: "vec_norm v * vec_norm w * (v ∙c (1 / vec_norm w ⋅⇩v w)) = 0" by auto
    moreover have "v ∙c w = 0" using asm dim0 by auto
    ultimately have "v ∙c w = vec_norm v * vec_norm w * (v ∙c (1 / vec_norm w ⋅⇩v w))" by auto
  }
  with dim show "w ≠ 0⇩v (dim_vec w) ⟹ v = 0⇩v (dim_vec v) ⟹ v ∙c w = vec_norm v * vec_norm w * (v ∙c (1 / vec_norm w ⋅⇩v w))" by auto
  {
    assume asmw: "w ≠ 0⇩v n" and asmv: "v ≠ 0⇩v n"
    have "vec_norm w > 0" by (insert asmw dim0, rule vec_norm_ge_0, auto)
    then have cw: "conjugate (1 / vec_norm w) = 1 / vec_norm w"
      by (simp add: complex_eq_iff complex_is_Real_iff less_complex_def) 
    from dim0 have 
      "((1 / vec_norm v ⋅⇩v v) ∙c (1 / vec_norm w ⋅⇩v w)) = 1 / vec_norm v * (v ∙c (1 / vec_norm w ⋅⇩v w))" by auto
    also have "… = 1 / vec_norm v * (v ∙ (conjugate (1 / vec_norm w) ⋅⇩v conjugate w))"
      by (subst conjugate_smult_vec, auto)
    also have "… = 1 / vec_norm v * conjugate (1 / vec_norm w) * (v ∙ conjugate w)" using dim by auto
    also have "… = 1 / vec_norm v * (1 / vec_norm w) * (v ∙c w)" using vec_norm_ge_0 cw by auto
    finally have eq1: "(1 / vec_norm v ⋅⇩v v) ∙c (1 / vec_norm w ⋅⇩v w) = 1 / vec_norm v * (1 / vec_norm w) * (v ∙c w)" .
    then have "vec_norm v * vec_norm w * ((1 / vec_norm v ⋅⇩v v) ∙c (1 / vec_norm w ⋅⇩v w)) = (v ∙c w)" 
      by (subst eq1, insert vec_norm_ge_0[of v n, OF dim_v asmv] vec_norm_ge_0[of w n, OF dim_w asmw], auto)
  }
  with dim show " w ≠ 0⇩v (dim_vec w) ⟹ v ≠ 0⇩v (dim_vec v) ⟹ v ∙c w = vec_norm v * vec_norm w * ((1 / vec_norm v ⋅⇩v v) ∙c (1 / vec_norm w ⋅⇩v w))" by auto
qed

lemma normalized_vec_norm :
  fixes v :: "complex vec"
  assumes dim_v: "v ∈ carrier_vec n" 
    and neq0: "v ≠ 0⇩v n"
  shows "vec_normalize v ∙c vec_normalize v = 1"
  unfolding vec_normalize_def
proof (simp, rule conjI)
  show "v = 0⇩v (dim_vec v) ⟶ v ∙c v = 1" using neq0 dim_v by auto
  have dim_a: "(vec_normalize v) ∈ carrier_vec n" "conjugate (vec_normalize v) ∈ carrier_vec n" using dim_v vec_normalize_def by auto 
  note dim = dim_v dim_a
  have nvge0: "vec_norm v > 0" using vec_norm_ge_0 neq0 dim_v by auto
  then have vvvv: "v ∙c v = (vec_norm v) * (vec_norm v)" unfolding vec_norm_def by (metis power2_csqrt power2_eq_square)
  from nvge0 have "conjugate (vec_norm v) = vec_norm v"
    by (simp add: complex_eq_iff complex_is_Real_iff less_complex_def) 
  then have "v ∙c (1 / vec_norm v ⋅⇩v v) = 1 / vec_norm v * (v ∙c v)" 
    by (subst conjugate_smult_vec, auto)
  also have "… = 1 / vec_norm v * vec_norm v * vec_norm v" using vvvv by auto
  also have "… = vec_norm v" by auto
  finally have "v ∙c (1 / vec_norm v ⋅⇩v v) = vec_norm v".
  then show "v ≠ 0⇩v (dim_vec v) ⟶ vec_norm v ≠ 0 ∧ v ∙c (1 / vec_norm v ⋅⇩v v) = vec_norm v" 
    using neq0 nvge0 by auto
qed

lemma normalize_zero:
  assumes "v ∈ carrier_vec n"
  shows "vec_normalize v = 0⇩v n ⟷ v = 0⇩v n"
proof
  show "v = 0⇩v n ⟹ vec_normalize v = 0⇩v n" unfolding vec_normalize_def by auto
next
  have "v ≠ 0⇩v n ⟹ vec_normalize v ≠ 0⇩v n" unfolding vec_normalize_def 
  proof (simp, rule impI)
    assume asm: "v ≠ 0⇩v n"
    then have "vec_norm v > 0" using vec_norm_ge_0 assms by auto
    then have nvge0: "1 / vec_norm v > 0" by (simp add: complex_is_Real_iff less_complex_def)
    have "∃k < n. v $ k ≠ 0" using asm assms by auto
    then obtain k where kn: "k < n" and  vkneq0: "v $ k ≠ 0" by auto
    then have "(1 / vec_norm v ⋅⇩v v) $ k = (1 / vec_norm v) * (v $ k)" 
      using assms carrier_vecD index_smult_vec(1) by blast
    with nvge0 vkneq0 have "(1 / vec_norm v ⋅⇩v v) $ k ≠ 0" by auto
    then show "1 / vec_norm v ⋅⇩v v ≠ 0⇩v n" using assms kn by fastforce
  qed
  then show "vec_normalize v = 0⇩v n ⟹ v = 0⇩v n" by auto
qed

lemma normalize_normalize[simp]:
  "vec_normalize (vec_normalize v) = vec_normalize v"
proof (rule disjE[of "v = 0⇩v (dim_vec v)" "v ≠ 0⇩v (dim_vec v)"], auto)
  let ?n = "dim_vec v"
{
  assume "v = 0⇩v ?n"
  then have "vec_normalize v = v" unfolding vec_normalize_def by auto
  then show "vec_normalize (vec_normalize v) = vec_normalize v" by auto
}
  assume neq0: "v ≠ 0⇩v ?n"
  have dim: "v ∈ carrier_vec ?n" by auto
  have "vec_norm (vec_normalize v) = 1" unfolding vec_norm_def
    using normalized_vec_norm[OF dim neq0] by auto
  then show "vec_normalize (vec_normalize v) = vec_normalize v" 
    by (subst (1) vec_normalize_def, simp)
qed

subsection ‹Spectral decomposition of normal complex matrices›

lemma normalize_keep_corthogonal:
  fixes vs :: "complex vec list"
  assumes cor: "corthogonal vs" and dims: "set vs ⊆ carrier_vec n"
  shows "corthogonal (map vec_normalize vs)"
  unfolding corthogonal_def
proof (rule allI, rule impI, rule allI, rule impI, goal_cases)
  case c: (1 i j)
  let ?m = "length vs"
  have len: "length (map vec_normalize vs) = ?m" by auto
  have dim: "⋀k. k < ?m ⟹ (vs ! k) ∈ carrier_vec n" using dims by auto
  have map: "⋀k. k < ?m ⟹  map vec_normalize vs ! k = vec_normalize (vs ! k)" by auto

  have eq1: "⋀j k. j < ?m ⟹ k < ?m ⟹ ((vs ! j) ∙c (vs ! k) = 0) = (j ≠ k)" using assms unfolding corthogonal_def by auto
  then have "⋀k. k < ?m ⟹ (vs ! k) ∙c (vs ! k) ≠ 0 " by auto
  then have "⋀k. k < ?m ⟹ (vs ! k) ≠ (0⇩v n)" using dim 
    by (auto simp add: conjugate_square_eq_0_vec[of _ n, OF dim])
  then have vnneq0: "⋀k. k < ?m ⟹ vec_norm (vs ! k) ≠ 0" using vec_norm_zero[OF dim] by auto
  then have i0: "vec_norm (vs ! i) ≠ 0" and j0: "vec_norm (vs ! j) ≠ 0" using c by auto
  have "(vs ! i) ∙c (vs ! j) = vec_norm (vs ! i) * vec_norm (vs ! j) * (vec_normalize (vs ! i) ∙c vec_normalize (vs ! j))"
    by (subst normalized_cscalar_prod[of "vs ! i" n "vs ! j"], auto, insert dim c, auto)
  with i0 j0 have "(vec_normalize (vs ! i) ∙c vec_normalize (vs ! j) = 0) = ((vs ! i) ∙c (vs ! j) = 0)" by auto
  with eq1 c have "(vec_normalize (vs ! i) ∙c vec_normalize (vs ! j) = 0) = (i ≠ j)" by auto
  with map c show "(map vec_normalize vs ! i ∙c map vec_normalize vs ! j = 0) = (i ≠ j)" by auto
qed

lemma normalized_corthogonal_mat_is_unitary:
  assumes W: "set ws ⊆ carrier_vec n"
    and orth: "corthogonal ws"
    and len: "length ws = n"
  shows "unitary (mat_of_cols n (map vec_normalize ws))" (is "unitary ?W")
proof -
  define vs where "vs = map vec_normalize ws"
  define W where "W = mat_of_cols n vs"
  have W': "set vs ⊆ carrier_vec n" using assms vs_def by auto
  then have W'': "⋀k. k < length vs ⟹ vs ! k ∈ carrier_vec n" by auto
  have orth': "corthogonal vs" using assms normalize_keep_corthogonal vs_def by auto
  have len'[simp]: "length vs = n" using assms vs_def by auto
  have dimW: "W ∈ carrier_mat n n" using W_def len by auto
  have "adjoint W ∈ carrier_mat n n" using dimW by auto
  then have dimaW: "mat_adjoint W ∈ carrier_mat n n" by auto
  {
    fix i j assume i: "i < n" and j: "j < n"
    have dimws: "(ws ! i) ∈ carrier_vec n" "(ws ! j) ∈ carrier_vec n" using W len i j by auto
    have "(ws ! i) ∙c (ws ! i) ≠ 0" "(ws ! j) ∙c (ws ! j) ≠ 0" using orth corthogonal_def[of ws] len i j by auto
    then have neq0: "(ws ! i) ≠ 0⇩v n" "(ws ! j) ≠ 0⇩v n"
      by (auto simp add: conjugate_square_eq_0_vec[of "ws ! i" n])
    then have "vec_norm (ws ! i) > 0" "vec_norm (ws ! j) > 0" using vec_norm_ge_0 dimws by auto
    then have ge0: "vec_norm (ws ! i) * vec_norm (ws ! j) > 0" by (auto simp: less_complex_def)
    have ws': "vs ! i = vec_normalize (ws ! i)" 
        "vs ! j = vec_normalize (ws ! j)" 
      using len i j vs_def by auto
    have ii1: "(vs ! i) ∙c (vs ! i) = 1" 
      apply (simp add: ws')
      apply (rule normalized_vec_norm[of "ws ! i"], rule dimws, rule neq0)
      done
    have ij0: "i ≠ j ⟹ (ws ! i)  ∙c (ws ! j) = 0" using i j 
      by (insert orth, auto simp add: corthogonal_def[of ws] len)
    have "i ≠ j ⟹ (ws ! i)  ∙c (ws ! j) =  (vec_norm (ws ! i) * vec_norm (ws ! j)) * ((vs ! i) ∙c (vs ! j))"
      apply (auto simp add: ws')
      apply (rule normalized_cscalar_prod)
       apply (rule dimws, rule dimws)
      done
    with ij0 have ij0': "i ≠ j ⟹ (vs ! i) ∙c (vs ! j) = 0" using ge0 by auto
    have cWk: "⋀k. k < n ⟹ col W k = vs ! k" unfolding W_def 
    apply (subst col_mat_of_cols)
      apply (auto simp add: W'')
      done
    have "(mat_adjoint W * W) $$ (j, i) = row (mat_adjoint W) j ∙ col W i"
      by (insert dimW i j dimaW, auto)
    also have "… = conjugate (col W j) ∙ col W i" 
      by (insert dimW i j dimaW, auto simp add: mat_adjoint_def)
    also have "… = col W i ∙ conjugate (col W j)" using comm_scalar_prod[of "col W i" n] dimW by auto
    also have "… = (vs ! i) ∙c (vs ! j)" using W_def col_mat_of_cols i j len cWk by auto
    finally have "(mat_adjoint W * W) $$ (j, i) = (vs ! i) ∙c (vs ! j)".
    then have "(mat_adjoint W * W) $$ (j, i) = (if (j = i) then 1 else 0)"
      by (auto simp add: ii1 ij0')
  }
  note maWW = this
  then have "mat_adjoint W * W = 1⇩m n" unfolding one_mat_def using dimW dimaW
    by (auto simp add: maWW adjoint_def)
  then have iv0: "adjoint W * W = 1⇩m n"  by auto
  have dimaW: "adjoint W ∈ carrier_mat n n" using dimaW by auto
  then have iv1: "W * adjoint W  = 1⇩m n" using mat_mult_left_right_inverse dimW iv0 by auto
  then show "unitary W" unfolding unitary_def inverts_mat_def using dimW dimaW iv0 iv1 by auto 
qed

lemma normalize_keep_eigenvector:
  assumes ev: "eigenvector A v e" 
    and dim: "A ∈ carrier_mat n n" "v ∈ carrier_vec n"
  shows "eigenvector A (vec_normalize v) e"
  unfolding eigenvector_def
proof 
  show "vec_normalize v ∈ carrier_vec (dim_row A)" using dim by auto
  have eg: "A *⇩v v = e ⋅⇩v v" using ev dim eigenvector_def by auto
  have vneq0: "v ≠ 0⇩v n" using ev dim unfolding eigenvector_def by auto
  then have s0: "vec_normalize v ≠ 0⇩v n" 
    by (insert dim, subst normalize_zero[of v], auto)
  from vneq0 have vvge0: "vec_norm v > 0" using vec_norm_ge_0 dim by auto
  have s1: "A *⇩v vec_normalize v = e ⋅⇩v vec_normalize v" unfolding vec_normalize_def 
    using vneq0 dim 
    apply (auto, simp add: mult_mat_vec)
    apply (subst eg, auto)
    done
  with s0 dim show "vec_normalize v ≠ 0⇩v (dim_row A) ∧ A *⇩v vec_normalize v = e ⋅⇩v vec_normalize v" by auto
qed

lemma four_block_mat_adjoint:
  fixes A B C D :: "'a::conjugatable_field mat"
  assumes dim: "A ∈ carrier_mat nr1 nc1" "B ∈ carrier_mat nr1 nc2"
  "C ∈ carrier_mat nr2 nc1" "D ∈ carrier_mat nr2 nc2"
  shows "adjoint (four_block_mat A B C D) 
    = four_block_mat (adjoint A) (adjoint C) (adjoint B) (adjoint D)"
  by (rule eq_matI, insert dim, auto simp add: adjoint_eval)

fun unitary_schur_decomposition :: "complex mat ⇒ complex list ⇒ complex mat × complex mat × complex mat" where 
  "unitary_schur_decomposition A [] = (A, 1⇩m (dim_row A), 1⇩m (dim_row A))"
| "unitary_schur_decomposition A (e # es) = (let
       n = dim_row A;
       n1 = n - 1;
       v' = find_eigenvector A e;
       v = vec_normalize v';
       ws0 = gram_schmidt n (basis_completion v);
       ws = map vec_normalize ws0;
       W = mat_of_cols n ws;
       W' = corthogonal_inv W;
       A' = W' * A * W;
       (A1,A2,A0,A3) = split_block A' 1 1;
       (B,P,Q) = unitary_schur_decomposition A3 es;
       z_row = (0⇩m 1 n1);
       z_col = (0⇩m n1 1);
       one_1 = 1⇩m 1
     in (four_block_mat A1 (A2 * P) A0 B, 
     W * four_block_mat one_1 z_row z_col P, 
     four_block_mat one_1 z_row z_col Q * W'))"

theorem unitary_schur_decomposition:
  assumes A: "(A::complex mat) ∈ carrier_mat n n"
      and c: "char_poly A = (∏ (e :: complex) ← es. [:- e, 1:])"
      and B: "unitary_schur_decomposition A es = (B,P,Q)"
  shows "similar_mat_wit A B P Q ∧ upper_triangular B ∧ diag_mat B = es ∧ unitary P ∧ (Q = adjoint P)"
  using assms
proof (induct es arbitrary: n A B P Q)
  case Nil
  with degree_monic_char_poly[of A n] 
  show ?case by (auto intro: similar_mat_wit_refl simp: diag_mat_def unitary_zero)
next
  case (Cons e es n A C P Q)
  let ?n1 = "n - 1"
  from Cons have A: "A ∈ carrier_mat n n" and dim: "dim_row A = n" by auto
  let ?cp = "char_poly A"
  from Cons(3)
  have cp: "?cp = [: -e, 1 :] * (∏e ← es. [:- e, 1:])" by auto
  have mon: "monic (∏e← es. [:- e, 1:])" by (rule monic_prod_list, auto)
  have deg: "degree ?cp = Suc (degree (∏e← es. [:- e, 1:]))" unfolding cp
    by (subst degree_mult_eq, insert mon, auto)
  with degree_monic_char_poly[OF A] have n: "n ≠ 0" by auto
  define v' where "v' = find_eigenvector A e"
  define v where "v = vec_normalize v'"
  define b where "b = basis_completion v"
  define ws0 where "ws0 = gram_schmidt n b"
  define ws where "ws = map vec_normalize ws0"
  define W where "W = mat_of_cols n ws"
  define W' where "W' = corthogonal_inv W"
  define A' where "A' = W' * A * W"
  obtain A1 A2 A0 A3 where splitA': "split_block A' 1 1 = (A1,A2,A0,A3)"
    by (cases "split_block A' 1 1", auto)
  obtain B P' Q' where schur: "unitary_schur_decomposition A3 es = (B,P',Q')" 
    by (cases "unitary_schur_decomposition A3 es", auto)
  let ?P' = "four_block_mat (1⇩m 1) (0⇩m 1 ?n1) (0⇩m ?n1 1) P'"
  let ?Q' = "four_block_mat (1⇩m 1) (0⇩m 1 ?n1) (0⇩m ?n1 1) Q'"
  have C: "C = four_block_mat A1 (A2 * P') A0 B" and P: "P = W * ?P'" and Q: "Q = ?Q' * W'"
    using Cons(4) unfolding unitary_schur_decomposition.simps
    Let_def list.sel dim
    v'_def[symmetric] v_def[symmetric] b_def[symmetric] ws0_def[symmetric] ws_def[symmetric] W'_def[symmetric] W_def[symmetric]
    A'_def[symmetric] split splitA' schur by auto
  have e: "eigenvalue A e" 
    unfolding eigenvalue_root_char_poly[OF A] cp by simp
  from find_eigenvector[OF A e] have ev': "eigenvector A v' e" unfolding v'_def .
  then have "v' ∈ carrier_vec n" unfolding eigenvector_def using A by auto
  with ev' have ev: "eigenvector A v e" unfolding v_def using A dim normalize_keep_eigenvector by auto
  from this[unfolded eigenvector_def]
  have v[simp]: "v ∈ carrier_vec n" and v0: "v ≠ 0⇩v n" using A by auto
  interpret cof_vec_space n "TYPE(complex)" .
  from basis_completion[OF v v0, folded b_def]
  have span_b: "span (set b) = carrier_vec n" and dist_b: "distinct b" 
    and indep: "¬ lin_dep (set b)" and b: "set b ⊆ carrier_vec n" and hdb: "hd b = v" 
    and len_b: "length b = n" by auto
  from hdb len_b n obtain vs where bv: "b = v # vs" by (cases b, auto)
  from gram_schmidt_result[OF b dist_b indep refl, folded ws0_def]
  have ws0: "set ws0 ⊆ carrier_vec n" "corthogonal ws0" "length ws0 = n" 
    by (auto simp: len_b)
  then have ws: "set ws ⊆ carrier_vec n" "corthogonal ws" "length ws = n" unfolding ws_def
    using normalize_keep_corthogonal by auto
  have ws0ne: "ws0 ≠ []" using ‹length ws0 = n› n by auto
  from gram_schmidt_hd[OF v, of vs, folded bv] have hdws0: "hd ws0 = (vec_normalize v')" unfolding ws0_def v_def .
  have "hd ws = vec_normalize (hd ws0)" unfolding ws_def using hd_map[OF ws0ne]  by auto
  then have hdws: "hd ws = v" unfolding v_def using normalize_normalize[of v'] hdws0 by auto
  have orth_W: "corthogonal_mat W" using orthogonal_mat_of_cols ws unfolding W_def.
  have W: "W ∈ carrier_mat n n"
    using ws unfolding W_def using mat_of_cols_carrier(1)[of n ws] by auto
  have W': "W' ∈ carrier_mat n n" unfolding W'_def corthogonal_inv_def using W 
    by (auto simp: mat_of_rows_def)  
  from corthogonal_inv_result[OF orth_W] 
  have W'W: "inverts_mat W' W" unfolding W'_def .
  hence WW': "inverts_mat W W'" using mat_mult_left_right_inverse[OF W' W] W' W
    unfolding inverts_mat_def by auto
  have A': "A' ∈ carrier_mat n n" using W W' A unfolding A'_def by auto
  have A'A_wit: "similar_mat_wit A' A W' W"
    by (rule similar_mat_witI[of _ _ n], insert W W' A A' W'W WW', auto simp: A'_def
    inverts_mat_def)
  hence A'A: "similar_mat A' A" unfolding similar_mat_def by blast
  from similar_mat_wit_sym[OF A'A_wit] have simAA': "similar_mat_wit A A' W W'" by auto
  have eigen[simp]: "A *⇩v v = e ⋅⇩v v" and v0: "v ≠ 0⇩v n"
    using v_def v'_def find_eigenvector[OF A e] A normalize_keep_eigenvector
    unfolding eigenvector_def by auto
  let ?f = "(λ i. if i = 0 then e else 0)"
  have col0: "col A' 0 = vec n ?f"
    unfolding A'_def W'_def W_def
    using corthogonal_col_ev_0[OF A v v0 eigen n hdws ws].
  from A' n have "dim_row A' = 1 + ?n1" "dim_col A' = 1 + ?n1" by auto
  from split_block[OF splitA' this] have A2: "A2 ∈ carrier_mat 1 ?n1"
    and A3: "A3 ∈ carrier_mat ?n1 ?n1" 
    and A'block: "A' = four_block_mat A1 A2 A0 A3" by auto
  have A1id: "A1 = mat 1 1 (λ _. e)"
    using splitA'[unfolded split_block_def Let_def] arg_cong[OF col0, of "λ v. v $ 0"] A' n
    by (auto simp: col_def)
  have A1: "A1 ∈ carrier_mat 1 1" unfolding A1id by auto
  {
    fix i
    assume "i < ?n1"
    with arg_cong[OF col0, of "λ v. v $ Suc i"] A'
    have "A' $$ (Suc i, 0) = 0" by auto
  } note A'0 = this
  have A0id: "A0 = 0⇩m ?n1 1"
    using splitA'[unfolded split_block_def Let_def] A'0 A' by auto
  have A0: "A0 ∈ carrier_mat ?n1 1" unfolding A0id by auto
  from cp char_poly_similar[OF A'A]
  have cp: "char_poly A' = [: -e,1 :] * (∏ e ← es. [:- e, 1:])" by simp
  also have "char_poly A' = char_poly A1 * char_poly A3" 
    unfolding A'block A0id
    by (rule char_poly_four_block_zeros_col[OF A1 A2 A3])
  also have "char_poly A1 = [: -e,1 :]"
    by (simp add: A1id char_poly_defs det_def)
  finally have cp: "char_poly A3 = (∏ e ← es. [:- e, 1:])"
    by (metis mult_cancel_left pCons_eq_0_iff zero_neq_one)
  from Cons(1)[OF A3 cp schur]
  have simIH: "similar_mat_wit A3 B P' Q'" and ut: "upper_triangular B" and diag: "diag_mat B = es"
    and uP': "unitary P'" and Q'P': "Q' = adjoint P'"
    by auto
  from similar_mat_witD2[OF A3 simIH] 
  have B: "B ∈ carrier_mat ?n1 ?n1" and P': "P' ∈ carrier_mat ?n1 ?n1" and Q': "Q' ∈ carrier_mat ?n1 ?n1" 
    and PQ': "P' * Q' = 1⇩m ?n1" by auto
  have A0_eq: "A0 = P' * A0 * 1⇩m 1" unfolding A0id using P' by auto
  have simA'C: "similar_mat_wit A' C ?P' ?Q'" unfolding A'block C
    by (rule similar_mat_wit_four_block[OF similar_mat_wit_refl[OF A1] simIH _ A0_eq A1 A3 A0],
    insert PQ' A2 P' Q', auto)
  have ut1: "upper_triangular A1" unfolding A1id by auto
  have ut: "upper_triangular C" unfolding C A0id
    by (intro upper_triangular_four_block[OF _ B ut1 ut], auto simp: A1id)
  from A1id have diagA1: "diag_mat A1 = [e]" unfolding diag_mat_def by auto
  from diag_four_block_mat[OF A1 B] have diag: "diag_mat C = e # es" unfolding diag diagA1 C by simp

  have aW: "adjoint W ∈ carrier_mat n n" using W by auto
  have aW': "adjoint W' ∈ carrier_mat n n" using W' by auto
  have "unitary W" using W_def ws_def ws0 normalized_corthogonal_mat_is_unitary by auto
  then have ivWaW: "inverts_mat W (adjoint W)" using unitary_def W aW by auto
  with WW' have W'aW: "W' = (adjoint W)" using inverts_mat_unique W W' aW by auto
  then have "adjoint W' = W" using adjoint_adjoint by auto
  with ivWaW have "inverts_mat W' (adjoint W')" using inverts_mat_symm W aW W'aW by auto
  then have "unitary W'" using unitary_def W' by auto

  have newP': "P' ∈ carrier_mat (n - Suc 0) (n - Suc 0)" using P' by auto
  have rl: "⋀ x1 x2 x3 x4 y1 y2 y3 y4 f. x1 = y1 ⟹ x2 = y2 ⟹ x3 = y3 ⟹ x4 = y4 ⟹ f x1 x2 x3 x4 = f y1 y2 y3 y4" by simp
  have Q'aP': "?Q' = adjoint ?P'"
    apply (subst four_block_mat_adjoint, auto simp add: newP')
    apply (rule rl[where f2 = four_block_mat])
       apply (auto simp add: eq_matI adjoint_eval Q'P')
    done
  have "adjoint P = adjoint ?P' * adjoint W"  using W newP' n
    apply (simp add: P)
    apply (subst adjoint_mult[of W, symmetric])
     apply (auto simp add: W P' carrier_matD[of W n n])
    done
  also have "… = ?Q' * W'" using Q'aP' W'aW by auto
  also have "… = Q" using Q by auto
  finally have QaP: "Q = adjoint P" ..

  from similar_mat_wit_trans[OF simAA' simA'C, folded P Q] have smw: "similar_mat_wit A C P Q" by blast
  then have dimP: "P ∈ carrier_mat n n" and dimQ: "Q ∈ carrier_mat n n" unfolding similar_mat_wit_def using A by auto
  from smw have "P * Q = 1⇩m n" unfolding similar_mat_wit_def using A  by auto
  then have "inverts_mat P Q" using inverts_mat_def dimP by auto
  then have uP: "unitary P" using QaP unitary_def dimP by auto
    
  from ut similar_mat_wit_trans[OF simAA' simA'C, folded P Q] diag uP QaP
  show ?case by blast
qed

lemma complex_mat_char_poly_factorizable:
  fixes A :: "complex mat"
  assumes "A ∈ carrier_mat n n"
  shows "∃as. char_poly A =  (∏ a ← as. [:- a, 1:]) ∧ length as = n"
proof -
  let ?ca = "char_poly A"
  have ex0: "∃bs. Polynomial.smult (lead_coeff ?ca) (∏b←bs. [:- b, 1:]) = ?ca ∧
     length bs = degree ?ca"
    by (simp add: fundamental_theorem_algebra_factorized)
  then obtain bs where " Polynomial.smult (lead_coeff ?ca) (∏b←bs. [:- b, 1:]) = ?ca ∧
     length bs = degree ?ca" by auto
  moreover have "lead_coeff ?ca = (1::complex)" 
    using assms degree_monic_char_poly by blast
  ultimately have ex1: "?ca = (∏b←bs. [:- b, 1:]) ∧ length bs = degree ?ca" by auto
  moreover have "degree ?ca = n"
    by (simp add: assms degree_monic_char_poly)
  ultimately show ?thesis by auto
qed

lemma complex_mat_has_unitary_schur_decomposition:
  fixes A :: "complex mat"
  assumes "A ∈ carrier_mat n n"
  shows "∃B P es. similar_mat_wit A B P (adjoint P) ∧ unitary P 
    ∧ char_poly A = (∏ (e :: complex) ← es. [:- e, 1:]) ∧ diag_mat B = es"
proof -
  have "∃es. char_poly A =  (∏ e ← es. [:- e, 1:]) ∧ length es = n" 
    using assms by (simp add: complex_mat_char_poly_factorizable)
  then obtain es where es: "char_poly A =  (∏ e ← es. [:- e, 1:]) ∧ length es = n" by auto
  obtain B P Q where B: "unitary_schur_decomposition A es = (B,P,Q)" by (cases "unitary_schur_decomposition A es", auto)

  have "similar_mat_wit A B P Q ∧ upper_triangular B ∧ unitary P ∧ (Q = adjoint P) ∧ 
   char_poly A = (∏ (e :: complex) ← es. [:- e, 1:]) ∧ diag_mat B = es" using assms es B
    by (auto simp add: unitary_schur_decomposition)
  then show ?thesis by auto
qed

lemma normal_upper_triangular_matrix_is_diagonal:
  fixes A :: "'a::conjugatable_ordered_field mat"
  assumes "A ∈ carrier_mat n n"
    and tri: "upper_triangular A"
    and norm: "A * adjoint A = adjoint A * A"
  shows "diagonal_mat A"
proof (rule disjE[of "n = 0" "n > 0"], blast)
  have dim: "dim_row A = n" "dim_col A = n" using assms by auto
  from norm have eq0: "⋀i j. (A * adjoint A)$$(i,j) = (adjoint A * A)$$(i,j)" by auto
  have nat_induct_strong: 
    "⋀P. (P::nat⇒bool) 0 ⟹ (⋀i. i < n ⟹ (⋀k. k < i ⟹ P k) ⟹ P i) ⟹ (⋀i. i < n ⟹ P i)"
    by (metis dual_order.strict_trans infinite_descent0 linorder_neqE_nat)
  show "n = 0 ⟹ ?thesis" using dim unfolding diagonal_mat_def by auto
  show "n > 0 ⟹ ?thesis" unfolding diagonal_mat_def dim
    apply (rule allI, rule impI)
    apply (rule nat_induct_strong)
  proof (rule allI, rule impI, rule impI)
    assume asm: "n > 0"
    from tri upper_triangularD[of A 0 j] dim have z0: "⋀j. 0< j ⟹ j < n ⟹ A$$(j, 0) = 0" 
      by auto
    then have ada00: "(adjoint A * A)$$(0,0) = conjugate (A$$(0,0)) * A$$(0,0)"
      using asm dim by (auto simp add: scalar_prod_def adjoint_eval sum.atLeast_Suc_lessThan)
    have aad00: "(A * adjoint A)$$(0,0) = (∑k=0..<n. A$$(0, k) * conjugate (A$$(0, k)))"
      using asm dim by (auto simp add: scalar_prod_def adjoint_eval)
    moreover have 
      "… = A$$(0,0)