# Theory QHLProver.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"

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

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)"
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

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))"
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
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 (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 (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))"
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›

"A ∈ carrier_mat n n ⟹ adjoint A ∈ carrier_mat n n"

"adjoint A = mat (dim_col A) (dim_row A) (λ(i,j). conjugate (A \$\$ (j,i)))"

assumes "i < dim_col A" "j < dim_row A"
shows "(adjoint A) \$\$ (i,j) = conjugate (A \$\$ (j,i))"

assumes "i < dim_col A"
shows "row (adjoint A) i = conjugate (col A i)"
apply (rule eq_vecI)

assumes "i < dim_row A"
shows "col (adjoint A) i = conjugate (row A i)"
apply (rule eq_vecI)

text ‹The identity <v, A w> = <A* v, w>›
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))"
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

shows "adjoint (1⇩m n) = (1⇩m n::complex mat)"
apply (rule eq_matI)

fixes A :: "'a::conjugatable_field mat"
apply (rule eq_matI) using conjugatable_ring_class.conjugate_dist_mul

fixes A B :: "'a::conjugatable_field mat"
assumes "A ∈ carrier_mat n m" "B ∈ carrier_mat n m"
apply (rule eq_matI)

fixes A B :: "'a::conjugatable_field mat"
assumes "A ∈ carrier_mat n m" "B ∈ carrier_mat n m"
apply (rule eq_matI)

fixes A B :: "'a::conjugatable_field mat"
assumes "A ∈ carrier_mat n m" "B ∈ carrier_mat m l"
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"])
qed

fixes A :: "'a::conjugatable_field mat"

fixes A :: "complex mat"
shows "trace (A * adjoint A) ≥ 0"
apply (rule sum_nonneg) by auto

subsection ‹Algebraic manipulations on matrices›

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

"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"

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"
"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)"
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)

assumes "A ∈ carrier_mat n n" "unitary A"
unfolding unitary_def

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)"
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 (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 (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)
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"
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
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 (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

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)

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
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 (rule rl[where f2 = four_block_mat])
done
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"
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"
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
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"
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