# Theory Perron_Frobenius_Irreducible

theory Perron_Frobenius_Irreducible
imports Perron_Frobenius Roots_Unity Miscellaneous
```(* Author: Thiemann *)
subsection ‹The Perron Frobenius Theorem for Irreducible Matrices›

theory Perron_Frobenius_Irreducible
imports
Perron_Frobenius
Roots_Unity
Rank_Nullity_Theorem.Miscellaneous (* for scalar-matrix-multiplication,
this import is incompatible with field_simps, ac_simps *)
begin

lifting_forget vec.lifting
lifting_forget mat.lifting
lifting_forget poly.lifting

lemma charpoly_of_real: "charpoly (map_matrix complex_of_real A) = map_poly of_real (charpoly A)"
by (transfer_hma rule: of_real_hom.char_poly_hom)

context includes lifting_syntax
begin
lemma HMA_M_smult[transfer_rule]: "((=) ===> HMA_M ===> HMA_M) (⋅⇩m) ((*k))"
unfolding smult_mat_def
unfolding rel_fun_def HMA_M_def from_hma⇩m_def
by (auto simp: matrix_scalar_mult_def)
end

lemma order_charpoly_smult: fixes A :: "complex ^ 'n ^ 'n"
assumes k: "k ≠ 0"
shows "order x (charpoly (k *k A)) = order (x / k) (charpoly A)"
by (transfer fixing: k, rule order_char_poly_smult[OF _ k])

(* use field, since the *k-lemmas have been stated for fields *)
lemma smult_eigen_vector: fixes a :: "'a :: field"
assumes "eigen_vector A v x"
shows "eigen_vector (a *k A) v (a * x)"
proof -
from assms[unfolded eigen_vector_def] have v: "v ≠ 0" and id: "A *v v = x *s v" by auto
from arg_cong[OF id, of "(*s) a"] have id: "(a *k A) *v v = (a * x) *s v"
unfolding scalar_matrix_vector_assoc by simp
thus "eigen_vector (a *k A) v (a * x)" using v unfolding eigen_vector_def by auto
qed

lemma smult_eigen_value: fixes a :: "'a :: field"
assumes "eigen_value A x"
shows "eigen_value (a *k A) (a * x)"
using assms smult_eigen_vector[of A _ x a] unfolding eigen_value_def by blast

locale fixed_mat = fixes A :: "'a :: zero ^ 'n ^ 'n"
begin
definition G :: "'n rel" where
"G = { (i,j). A \$ i \$ j ≠ 0}"

definition irreducible :: bool where
"irreducible = (UNIV ⊆ G^+)"
end

lemma G_transpose:
"fixed_mat.G (transpose A) = ((fixed_mat.G A))^-1"
unfolding fixed_mat.G_def by (force simp: transpose_def)

lemma G_transpose_trancl:
"(fixed_mat.G (transpose A))^+ = ((fixed_mat.G A)^+)^-1"
unfolding G_transpose trancl_converse by auto

locale pf_nonneg_mat = fixed_mat A for
A :: "'a :: linordered_idom ^ 'n ^ 'n" +
assumes non_neg_mat: "non_neg_mat A"
begin
lemma nonneg: "A \$ i \$ j ≥ 0"
using non_neg_mat unfolding non_neg_mat_def elements_mat_h_def by auto

lemma nonneg_matpow: "matpow A n \$ i \$ j ≥ 0"
by (induct n arbitrary: i j, insert nonneg,
auto intro!: sum_nonneg simp: matrix_matrix_mult_def mat_def)

lemma G_relpow_matpow_pos: "(i,j) ∈ G ^^ n ⟹ matpow A n \$ i \$ j > 0"
proof (induct n arbitrary: i j)
case (0 i)
thus ?case by (auto simp: mat_def)
next
case (Suc n i j)
from Suc(2) have "(i,j) ∈ G ^^ n O G"
then obtain k where
ik: "A \$ k \$ j ≠ 0" and kj: "(i, k) ∈ G ^^ n" by (auto simp: G_def)
from ik nonneg[of k j] have ik: "A \$ k \$ j > 0" by auto
from Suc(1)[OF kj] have IH: "matpow A n \$h i \$h k > 0" .
thus ?case using ik by (auto simp: nonneg_matpow nonneg matrix_matrix_mult_def
intro!: sum_pos2[of _ k] mult_nonneg_nonneg)
qed

lemma matpow_mono: assumes B: "⋀ i j. B \$ i \$ j ≥ A \$ i \$ j"
shows "matpow B n \$ i \$ j ≥ matpow A n \$ i \$ j"
proof (induct n arbitrary: i j)
case (Suc n i j)
thus ?case using B nonneg_matpow[of n] nonneg
by (auto simp: matrix_matrix_mult_def intro!: sum_mono mult_mono')
qed simp

lemma matpow_sum_one_mono: "matpow (A + mat 1) (n + k) \$ i \$ j ≥ matpow (A + mat 1) n \$ i \$ j"
proof (induct k)
case (Suc k)
have "(matpow (A + mat 1) (n + k) ** A) \$h i \$h j ≥ 0" unfolding matrix_matrix_mult_def
using order.trans[OF nonneg_matpow matpow_mono[of "A + mat 1" "n + k"]]
by (auto intro!: sum_nonneg mult_nonneg_nonneg nonneg simp: mat_def)
qed simp

lemma G_relpow_matpow_pos_ge:
assumes "(i,j) ∈ G ^^ m" "n ≥ m"
shows "matpow (A + mat 1) n \$ i \$ j > 0"
proof -
from assms(2) obtain k where n: "n = m + k" using le_Suc_ex by blast
have "0 < matpow A m \$ i \$ j" by (rule G_relpow_matpow_pos[OF assms(1)])
also have "… ≤ matpow (A + mat 1) m \$ i \$ j"
by (rule matpow_mono, auto simp: mat_def)
also have "… ≤ matpow (A + mat 1) n \$ i \$ j" unfolding n using matpow_sum_one_mono .
finally show ?thesis .
qed
end

locale perron_frobenius = pf_nonneg_mat A
for A :: "real ^ 'n ^ 'n" +
assumes irr: irreducible
begin

definition N where "N = (SOME N. ∀ ij. ∃ n ≤ N. ij ∈ G ^^ n)"

lemma N: "∃ n ≤ N. ij ∈ G ^^ n"
proof -
{
fix ij
have "ij ∈ G^+" using irr[unfolded irreducible_def] by auto
from this[unfolded trancl_power] have "∃ n. ij ∈ G ^^ n" by blast
}
hence "∀ ij. ∃ n. ij ∈ G ^^ n" by auto
from choice[OF this] obtain f where f: "⋀ ij. ij ∈ G ^^ (f ij)" by auto
define N where N: "N = Max (range f)"
{
fix ij
from f[of ij] have "ij ∈ G ^^ f ij" .
moreover have "f ij ≤ N" unfolding N
by (rule Max_ge, auto)
ultimately have "∃ n ≤ N. ij ∈ G ^^ n" by blast
} note main = this
let ?P = "λ N. ∀ ij. ∃ n ≤ N. ij ∈ G ^^ n"
from main have "?P N" by blast
from someI[of ?P, OF this, folded N_def]
show ?thesis by blast
qed

lemma irreducible_matpow_pos: assumes irreducible
shows "matpow (A + mat 1) N \$ i \$ j > 0"
proof -
from N obtain n where n: "n ≤ N" and reach: "(i,j) ∈ G ^^ n" by auto
show ?thesis by (rule G_relpow_matpow_pos_ge[OF reach n])
qed

lemma pf_transpose: "perron_frobenius (transpose A)"
proof
show "fixed_mat.irreducible (transpose A)"
unfolding fixed_mat.irreducible_def G_transpose_trancl using irr[unfolded irreducible_def]
by auto
qed (insert nonneg, auto simp: transpose_def non_neg_mat_def elements_mat_h_def)

abbreviation le_vec :: "real ^ 'n ⇒ real ^ 'n ⇒ bool" where
"le_vec x y ≡ (∀ i. x \$ i ≤ y \$ i)"

abbreviation lt_vec :: "real ^ 'n ⇒ real ^ 'n ⇒ bool" where
"lt_vec x y ≡ (∀ i. x \$ i < y \$ i)"

definition "A1n = matpow (A + mat 1) N"

lemmas A1n_pos = irreducible_matpow_pos[OF irr, folded A1n_def]

definition r :: "real ^ 'n ⇒ real" where
"r x = Min { (A *v x) \$ j / x \$ j | j. x \$ j ≠ 0 }"

definition X :: "(real ^ 'n )set" where
"X = { x . le_vec 0 x ∧ x ≠ 0 }"

lemma nonneg_Ax: "x ∈ X ⟹ le_vec 0 (A *v x)"
unfolding X_def using nonneg
by (auto simp: matrix_vector_mult_def intro!: sum_nonneg)

lemma A_nonzero_fixed_i: "∃ j. A \$ i \$ j ≠ 0"
proof -
from irr[unfolded irreducible_def] have "(i,i) ∈ G^+" by auto
then obtain j where "(i,j) ∈ G" by (metis converse_tranclE)
hence Aij: "A \$ i \$ j ≠ 0" unfolding G_def by auto
thus ?thesis ..
qed

lemma A_nonzero_fixed_j: "∃ i. A \$ i \$ j ≠ 0"
proof -
from irr[unfolded irreducible_def] have "(j,j) ∈ G^+" by auto
then obtain i where "(i,j) ∈ G" by (cases, auto)
hence Aij: "A \$ i \$ j ≠ 0" unfolding G_def by auto
thus ?thesis ..
qed

lemma Ax_pos: assumes x: "lt_vec 0 x"
shows "lt_vec 0 (A *v x)"
proof
fix i
from A_nonzero_fixed_i[of i] obtain j where "A \$ i \$ j ≠ 0" by auto
with nonneg[of i j] have A: "A \$ i \$ j > 0" by simp
from x have "x \$ j ≥ 0" for j by (auto simp: order.strict_iff_order)
note nonneg = mult_nonneg_nonneg[OF nonneg[of i] this]
have "(A *v x) \$ i = (∑j∈UNIV. A \$ i \$ j * x \$ j)"
unfolding matrix_vector_mult_def by simp
also have "… = A \$ i \$ j * x \$ j + (∑j∈UNIV - {j}. A \$ i \$ j * x \$ j)"
by (subst sum.remove, auto)
also have "… > 0 + 0"
by (rule add_less_le_mono, insert A x[rule_format] nonneg,
auto intro!: sum_nonneg mult_pos_pos)
finally show "0 \$ i < (A *v x) \$ i" by simp
qed

lemma nonzero_Ax: assumes x: "x ∈ X"
shows "A *v x ≠ 0"
proof
assume 0: "A *v x = 0"
from x[unfolded X_def] have x: "le_vec 0 x" "x ≠ 0" by auto
from x(2) obtain j where xj: "x \$ j ≠ 0"
by (metis vec_eq_iff zero_index)
from A_nonzero_fixed_j[of j]  obtain i where Aij: "A \$ i \$ j ≠ 0" by auto
from arg_cong[OF 0, of "λ v. v \$ i", unfolded matrix_vector_mult_def]
have "0 = (∑ k ∈ UNIV. A \$h i \$h k * x \$h k)" by auto
also have "… = A \$h i \$h j * x \$h j + (∑ k ∈ UNIV - {j}. A \$h i \$h k * x \$h k)"
by (subst sum.remove[of _ j], auto)
also have "… > 0 + 0"
by (rule add_less_le_mono, insert nonneg[of i] Aij x(1) xj,
auto intro!: sum_nonneg mult_pos_pos simp: dual_order.not_eq_order_implies_strict)
finally show False by simp
qed

lemma r_witness: assumes x: "x ∈ X"
shows "∃ j. x \$ j > 0 ∧ r x = (A *v x) \$ j / x \$ j"
proof -
from x[unfolded X_def] have x: "le_vec 0 x" "x ≠ 0" by auto
let ?A = "{ (A *v x) \$ j / x \$ j | j. x \$ j ≠ 0 }"
from x(2) obtain j where "x \$ j ≠ 0"
by (metis vec_eq_iff zero_index)
hence empty: "?A ≠ {}" by auto
from Min_in[OF _ this, folded r_def]
obtain j where "x \$ j ≠ 0" and rx: "r x = (A *v x) \$ j / x \$ j" by auto
with x have "x \$ j > 0" by (auto simp: dual_order.not_eq_order_implies_strict)
with rx show ?thesis by auto
qed

lemma rx_nonneg: assumes x: "x ∈ X"
shows "r x ≥ 0"
proof -
from x[unfolded X_def] have x: "le_vec 0 x" "x ≠ 0" by auto
let ?A = "{ (A *v x) \$ j / x \$ j | j. x \$ j ≠ 0 }"
from r_witness[OF ‹x ∈ X›]
have empty: "?A ≠ {}" by force
show ?thesis unfolding r_def X_def
proof (subst Min_ge_iff, force, use empty in force, intro ballI)
fix y
assume "y ∈ ?A"
then obtain j where "y = (A *v x) \$ j / x \$ j" and "x \$ j ≠ 0" by auto
from nonneg_Ax[OF ‹x ∈ X›] this x
show "0 ≤ y" by simp
qed
qed

lemma rx_pos: assumes x: "lt_vec 0 x"
shows "r x > 0"
proof -
from Ax_pos[OF x] have lt: "lt_vec 0 (A *v x)" .
from x have x': "x ∈ X" unfolding X_def order.strict_iff_order by auto
let ?A = "{ (A *v x) \$ j / x \$ j | j. x \$ j ≠ 0 }"
from r_witness[OF ‹x ∈ X›]
have empty: "?A ≠ {}" by force
show ?thesis unfolding r_def X_def
proof (subst Min_gr_iff, force, use empty in force, intro ballI)
fix y
assume "y ∈ ?A"
then obtain j where "y = (A *v x) \$ j / x \$ j" and "x \$ j ≠ 0" by auto
from lt this x show "0 < y" by simp
qed
qed

lemma rx_le_Ax: assumes x: "x ∈ X"
shows "le_vec (r x *s x) (A *v x)"
proof (intro allI)
fix i
show "(r x *s x) \$h i ≤ (A *v x) \$h i"
proof (cases "x \$ i = 0")
case True
with nonneg_Ax[OF x] show ?thesis by auto
next
case False
with x[unfolded X_def] have pos: "x \$ i > 0"
by (auto simp: dual_order.not_eq_order_implies_strict)
from False have "(A *v x) \$h i / x \$ i ∈ { (A *v x) \$ j / x \$ j | j. x \$ j ≠ 0 }" by auto
hence "(A *v x) \$h i / x \$ i ≥ r x" unfolding r_def by simp
hence "x \$ i * r x ≤ x \$ i * ((A *v x) \$h i / x \$ i)" unfolding mult_le_cancel_left_pos[OF pos] .
also have "… = (A *v x) \$h i" using pos by simp
finally show ?thesis by (simp add: ac_simps)
qed
qed

lemma rho_le_x_Ax_imp_rho_le_rx: assumes x: "x ∈ X"
and ρ: "le_vec (ρ *s x) (A *v x)"
shows "ρ ≤ r x"
proof -
from r_witness[OF x] obtain j where
rx: "r x = (A *v x) \$ j / x \$ j" and xj: "x \$ j > 0" "x \$ j ≥ 0" by auto
from divide_right_mono[OF ρ[rule_format, of j] xj(2)]
show ?thesis unfolding rx using xj by simp
qed

lemma rx_Max: assumes x: "x ∈ X"
shows "r x = Sup { ρ . le_vec (ρ *s x) (A *v x) }" (is "_ = Sup ?S")
proof -
have "r x ∈ ?S" using rx_le_Ax[OF x] by auto
moreover {
fix y
assume "y ∈ ?S"
hence y: "le_vec (y *s x) (A *v x)" by auto
from rho_le_x_Ax_imp_rho_le_rx[OF x this]
have "y ≤ r x" .
}
ultimately show ?thesis by (metis (mono_tags, lifting) cSup_eq_maximum)
qed

lemma r_smult: assumes x: "x ∈ X"
and a: "a > 0"
shows "r (a *s x) = r x"
unfolding r_def
by (rule arg_cong[of _ _ Min], unfold vector_smult_distrib, insert a, simp)

definition "X1 = (X ∩ {x. norm x = 1})"

lemma bounded_X1: "bounded X1" unfolding bounded_iff X1_def by auto

lemma closed_X1: "closed X1"
proof -
have X1: "X1 = {x. le_vec 0 x ∧ norm x = 1}"
unfolding X1_def X_def by auto
show ?thesis unfolding X1
by (intro closed_Collect_conj closed_Collect_all  closed_Collect_le closed_Collect_eq,
auto intro: continuous_intros)
qed

lemma compact_X1: "compact X1" using bounded_X1 closed_X1

definition "pow_A_1 x = A1n *v x"

lemma continuous_pow_A_1: "continuous_on R pow_A_1"
unfolding pow_A_1_def continuous_on
by (auto intro: tendsto_intros)

definition "Y = pow_A_1 ` X1"

lemma compact_Y: "compact Y"
unfolding Y_def using compact_X1 continuous_pow_A_1[of X1]
by (metis compact_continuous_image)

lemma Y_pos_main: assumes y: "y ∈ pow_A_1 ` X"
shows "y \$ i > 0"
proof -
from y obtain x where x: "x ∈ X" and y: "y = pow_A_1 x" unfolding Y_def X1_def by auto
from r_witness[OF x] obtain j where xj: "x \$ j > 0" by auto
from x[unfolded X_def] have xi: "x \$ i ≥ 0" for i by auto
have nonneg: "0 ≤ A1n \$ i \$ k * x \$ k" for k using A1n_pos[of i k] xi[of k] by auto
have "y \$ i = (∑j∈UNIV. A1n \$ i \$ j * x \$ j)"
unfolding y pow_A_1_def matrix_vector_mult_def by simp
also have "… = A1n \$ i \$ j * x \$ j + (∑j∈UNIV - {j}. A1n \$ i \$ j * x \$ j)"
by (subst sum.remove, auto)
also have "… > 0 + 0"
by (rule add_less_le_mono, insert xj A1n_pos nonneg,
auto intro!: sum_nonneg mult_pos_pos simp: dual_order.not_eq_order_implies_strict)
finally show ?thesis by simp
qed

lemma Y_pos: assumes y: "y ∈ Y"
shows "y \$ i > 0"
using Y_pos_main[of y i] y unfolding Y_def X1_def by auto

lemma Y_nonzero: assumes y: "y ∈ Y"
shows "y \$ i ≠ 0"
using Y_pos[OF y, of i] by auto

definition r' :: "real ^ 'n ⇒ real" where
"r' x = Min (range (λ j. (A *v x) \$ j / x \$ j))"

lemma r'_r: assumes x: "x ∈ Y" shows "r' x = r x"
unfolding r'_def r_def
proof (rule arg_cong[of _ _ Min])
have "range (λj. (A *v x) \$ j / x \$ j) ⊆ {(A *v x) \$ j / x \$ j |j. x \$ j ≠ 0}" (is "?L ⊆ ?R")
proof
fix y
assume "y ∈ ?L"
then obtain j where "y = (A *v x) \$ j / x \$ j" by auto
with Y_pos[OF x, of j] show "y ∈ ?R" by auto
qed
moreover have "?L ⊇ ?R" by auto
ultimately show "?L = ?R" by blast
qed

lemma continuous_Y_r: "continuous_on Y r"
proof -
have *: "(∀ y ∈ Y. P y (r y)) = (∀ y ∈ Y. P y (r' y))" for P using r'_r by auto
have "continuous_on Y r = continuous_on Y r'"
by (rule continuous_on_cong[OF refl r'_r[symmetric]])
also have …
unfolding continuous_on r'_def using Y_nonzero
by (auto intro!: tendsto_Min tendsto_intros)
finally show ?thesis .
qed

lemma X1_nonempty: "X1 ≠ {}"
proof -
define x where "x = ((χ i. if i = undefined then 1 else 0) :: real ^ 'n)"
{
assume "x = 0"
from arg_cong[OF this, of "λ x. x \$ undefined"] have False unfolding x_def by auto
}
hence x: "x ≠ 0" by auto
moreover have "le_vec 0 x" unfolding x_def by auto
moreover have "norm x = 1" unfolding norm_vec_def L2_set_def
by (auto, subst sum.remove[of _ undefined], auto simp: x_def)
ultimately show ?thesis unfolding X1_def X_def by auto
qed

lemma Y_nonempty: "Y ≠ {}"
unfolding Y_def using X1_nonempty by auto

definition z where "z = (SOME z. z ∈ Y ∧ (∀ y ∈ Y. r y ≤ r z))"

abbreviation "sr ≡ r z"

lemma z: "z ∈ Y" and sr_max_Y: "⋀ y. y ∈ Y ⟹ r y ≤ sr"
proof -
let ?P = "λ z. z ∈ Y ∧ (∀ y ∈ Y. r y ≤ r z)"
from continuous_attains_sup[OF compact_Y Y_nonempty continuous_Y_r]
obtain y where "?P y" by blast
from someI[of ?P, OF this, folded z_def]
show "z ∈ Y" "⋀ y. y ∈ Y ⟹ r y ≤ r z" by blast+
qed

lemma Y_subset_X: "Y ⊆ X"
proof
fix y
assume "y ∈ Y"
from Y_pos[OF this] show "y ∈ X" unfolding X_def
by (auto simp: order.strict_iff_order)
qed

lemma zX: "z ∈ X"
using z(1) Y_subset_X by auto

lemma le_vec_mono_left: assumes B: "⋀ i j. B \$ i \$ j ≥ 0"
and "le_vec x y"
shows "le_vec (B *v x) (B *v y)"
proof (intro allI)
fix i
show "(B *v x) \$ i ≤ (B *v y) \$ i" unfolding matrix_vector_mult_def using B[of i] assms(2)
by (auto intro!: sum_mono mult_left_mono)
qed

lemma matpow_1_commute: "matpow (A + mat 1) n ** A = A ** matpow (A + mat 1) n"
matrix_mul_assoc[symmetric])

lemma A1n_commute: "A1n ** A = A ** A1n"
unfolding A1n_def by (rule matpow_1_commute)

lemma le_vec_pow_A_1: assumes le: "le_vec (rho *s x) (A *v x)"
shows "le_vec (rho *s pow_A_1 x) (A *v pow_A_1 x)"
proof -
have "A1n \$ i \$ j ≥ 0" for i j using A1n_pos[of i j] by auto
from le_vec_mono_left[OF this le]
have "le_vec (A1n *v (rho *s x)) (A1n *v (A *v x))" .
also have "A1n *v (A *v x) = (A1n ** A) *v x" by (simp add: matrix_vector_mul_assoc)
also have "… = A *v (A1n *v x)" unfolding A1n_commute by (simp add: matrix_vector_mul_assoc)
also have "… = A *v (pow_A_1 x)" unfolding pow_A_1_def ..
also have "A1n *v (rho *s x) = rho *s (A1n *v x)" unfolding vector_smult_distrib ..
also have "… = rho *s pow_A_1 x" unfolding pow_A_1_def ..
finally show "le_vec (rho *s pow_A_1 x) (A *v pow_A_1 x)" .
qed

lemma r_pow_A_1: assumes x: "x ∈ X"
shows "r x ≤ r (pow_A_1 x)"
proof -
let ?y = "pow_A_1 x"
have "?y ∈ pow_A_1 ` X" using x by auto
from Y_pos_main[OF this]
have y: "?y ∈ X" unfolding X_def by (auto simp: order.strict_iff_order)
let ?A = "{ρ. le_vec (ρ *s x) (A *v x)}"
let ?B = "{ρ. le_vec (ρ *s pow_A_1 x) (A *v pow_A_1 x)}"
show ?thesis unfolding rx_Max[OF x] rx_Max[OF y]
proof (rule cSup_mono)
show "bdd_above ?B" using rho_le_x_Ax_imp_rho_le_rx[OF y] by fast
show "?A ≠ {}" using rx_le_Ax[OF x] by auto
fix rho
assume "rho ∈ ?A"
hence "le_vec (rho *s x) (A *v x)" by auto
from le_vec_pow_A_1[OF this] have "rho ∈ ?B" by auto
thus "∃ rho' ∈ ?B. rho ≤ rho'" by auto
qed
qed

lemma sr_max: assumes x: "x ∈ X"
shows "r x ≤ sr"
proof -
let ?n = "norm x"
define x' where "x' = inverse ?n *s x"
from x[unfolded X_def] have x0: "x ≠ 0" by auto
hence n: "?n > 0" by auto
have x': "x' ∈ X1" "x' ∈ X" using x n unfolding X1_def X_def x'_def by (auto simp: norm_smult)
have id: "r x = r x'" unfolding x'_def
by (rule sym, rule r_smult[OF x], insert n, auto)
define y where "y = pow_A_1 x'"
from x' have y: "y ∈ Y" unfolding Y_def y_def by auto
note id
also have "r x' ≤ r y" using r_pow_A_1[OF x'(2)] unfolding y_def .
also have "… ≤ r z" using sr_max_Y[OF y] .
finally show "r x ≤ r z" .
qed

lemma z_pos: "z \$ i > 0"
using Y_pos[OF z(1)] by auto

lemma sr_pos: "sr > 0"
by (rule rx_pos, insert z_pos, auto)

context fixes u
assumes u: "u ∈ X" and ru: "r u = sr"
begin

lemma sr_imp_eigen_vector_main: "sr *s u = A *v u"
proof (rule ccontr)
assume *: "sr *s u ≠ A *v u"
let ?x = "A *v u - sr *s u"
from * have 0: "?x ≠ 0" by auto
let ?y = "pow_A_1 u"
have "le_vec (sr *s u) (A *v u)" using rx_le_Ax[OF u] unfolding ru .
hence le: "le_vec 0 ?x" by auto
from 0 le have x: "?x ∈ X" unfolding X_def by auto
have y_pos: "lt_vec 0 ?y" using Y_pos_main[of ?y] u by auto
hence y: "?y ∈ X" unfolding X_def by (auto simp: order.strict_iff_order)
from Y_pos_main[of "pow_A_1 ?x"] x
have "lt_vec 0 (pow_A_1 ?x)" by auto
hence lt: "lt_vec (sr *s ?y) (A *v ?y)" unfolding pow_A_1_def matrix_vector_right_distrib_diff
matrix_vector_mul_assoc A1n_commute vector_smult_distrib by simp
let ?f = "(λ i. (A *v ?y - sr *s ?y) \$ i / ?y \$ i)"
let ?U = "UNIV :: 'n set"
define eps where "eps = Min (?f ` ?U)"
have U: "finite (?f ` ?U)" "?f ` ?U ≠ {}" by auto
have eps: "eps > 0" unfolding eps_def Min_gr_iff[OF U]
using lt sr_pos y_pos by auto
have le: "le_vec ((sr + eps) *s ?y) (A *v ?y)"
proof
fix i
have "((sr + eps) *s ?y) \$ i = sr * ?y \$ i + eps * ?y \$ i"
also have "… ≤ sr * ?y \$ i + ?f i * ?y \$ i"
show "0 ≤ ?y \$ i" using y_pos[rule_format, of i] by auto
show "eps ≤ ?f i" unfolding eps_def by (rule Min_le, auto)
qed
also have "… = (A *v ?y) \$ i" using sr_pos y_pos[rule_format, of i]
by simp
finally
show "((sr + eps) *s ?y) \$ i ≤ (A *v ?y) \$ i" .
qed
from rho_le_x_Ax_imp_rho_le_rx[OF y le]
have "r ?y ≥ sr + eps" .
with sr_max[OF y] eps show False by auto
qed

lemma sr_imp_eigen_vector: "eigen_vector A u sr"
unfolding eigen_vector_def sr_imp_eigen_vector_main using u unfolding X_def by auto

lemma sr_u_pos: "lt_vec 0 u"
proof -
let ?y = "pow_A_1 u"
define n where "n = N"
define c where "c = (sr + 1)^N"
have c: "c > 0" using sr_pos unfolding c_def by auto
have "lt_vec 0 ?y" using Y_pos_main[of ?y] u by auto
also have "?y = A1n *v u" unfolding pow_A_1_def ..
also have "… = c *s u" unfolding c_def A1n_def n_def[symmetric]
proof (induct n)
case (Suc n)
then show ?case
by (simp add: matrix_vector_mul_assoc[symmetric] algebra_simps vec.scale
sr_imp_eigen_vector_main[symmetric])
qed auto
finally have lt: "lt_vec 0 (c *s u)" .
have "0 < u \$ i" for i using lt[rule_format, of i] c by simp (metis zero_less_mult_pos)
thus "lt_vec 0 u" by simp
qed
end

lemma eigen_vector_z_sr: "eigen_vector A z sr"
using sr_imp_eigen_vector[OF zX refl] by auto

lemma eigen_value_sr: "eigen_value A sr"
using eigen_vector_z_sr unfolding eigen_value_def by auto

abbreviation "c ≡ complex_of_real"
abbreviation "cA ≡ map_matrix c A"
abbreviation "norm_v ≡ map_vector (norm :: complex ⇒ real)"

lemma norm_v_ge_0: "le_vec 0 (norm_v v)" by (auto simp: map_vector_def)
lemma norm_v_eq_0: "norm_v v = 0 ⟷ v = 0" by (auto simp: map_vector_def vec_eq_iff)

lemma cA_index: "cA \$ i \$ j = c (A \$ i \$ j)"
unfolding map_matrix_def map_vector_def by simp

lemma norm_cA[simp]: "norm (cA \$ i \$ j) = A \$ i \$ j"
using nonneg[of i j] by (simp add: cA_index)

context fixes α v
assumes ev: "eigen_vector cA v α"
begin

lemma evD: "α *s v = cA *v v" "v ≠ 0"
using ev[unfolded eigen_vector_def] by auto

lemma ev_alpha_norm_v: "norm_v (α *s v) = (norm α *s norm_v v)"
by (auto simp: map_vector_def norm_mult vec_eq_iff)

lemma ev_A_norm_v: "norm_v (cA *v v) \$ j ≤ (A *v norm_v v) \$ j"
proof -
have "norm_v (cA *v v) \$ j = norm (∑i∈UNIV. cA \$ j \$ i * v \$ i)"
unfolding map_vector_def by (simp add: matrix_vector_mult_def)
also have "… ≤ (∑i∈UNIV. norm (cA \$ j \$ i * v \$ i))" by (rule norm_sum)
also have "… = (∑i∈UNIV. A \$ j \$ i * norm_v v \$ i)"
by (rule sum.cong[OF refl], auto simp: norm_mult map_vector_def)
also have "… = (A *v norm_v v) \$ j" by (simp add: matrix_vector_mult_def)
finally show ?thesis .
qed

lemma ev_le_vec: "le_vec (norm α *s norm_v v) (A *v norm_v v)"
using arg_cong[OF evD(1), of norm_v, unfolded ev_alpha_norm_v] ev_A_norm_v by auto

lemma norm_v_X: "norm_v v ∈ X"
using norm_v_ge_0[of v] evD(2) norm_v_eq_0[of v] unfolding X_def by auto

lemma ev_inequalities: "norm α ≤ r (norm_v v)" "r (norm_v v) ≤ sr"
proof -
have v: "norm_v v ∈ X" by (rule norm_v_X)
from rho_le_x_Ax_imp_rho_le_rx[OF v ev_le_vec]
show "norm α ≤ r (norm_v v)" .
from sr_max[OF v]
show "r (norm_v v) ≤ sr" .
qed

lemma eigen_vector_norm_sr: "norm α ≤ sr" using ev_inequalities by auto
end

lemma eigen_value_norm_sr: assumes "eigen_value cA α"
shows "norm α ≤ sr"
using eigen_vector_norm_sr[of _ α] assms unfolding eigen_value_def by auto

lemma le_vec_trans: "le_vec x y ⟹ le_vec y u ⟹ le_vec x u"
using order.trans[of "x \$ i" "y \$ i" "u \$ i" for i] by auto

lemma eigen_vector_z_sr_c: "eigen_vector cA (map_vector c z) (c sr)"
unfolding of_real_hom.eigen_vector_hom by (rule eigen_vector_z_sr)

lemma eigen_value_sr_c: "eigen_value cA (c sr)"
using eigen_vector_z_sr_c unfolding eigen_value_def by auto

definition "w = perron_frobenius.z (transpose A)"

lemma w: "transpose A *v w = sr *s w" "lt_vec 0 w" "perron_frobenius.sr (transpose A) = sr"
proof -
interpret t: perron_frobenius "transpose A"
by (rule pf_transpose)
from eigen_vector_z_sr_c t.eigen_vector_z_sr_c
have ev: "eigen_value cA (c sr)" "eigen_value t.cA (c t.sr)"
unfolding eigen_value_def by auto
{
fix x
have "eigen_value (t.cA) x = eigen_value (transpose cA) x"
unfolding map_matrix_def map_vector_def transpose_def
by (auto simp: vec_eq_iff)
also have "… = eigen_value cA x" by (rule eigen_value_transpose)
finally have "eigen_value (t.cA) x = eigen_value cA x" .
} note ev_id = this
with ev have ev: "eigen_value t.cA (c sr)" "eigen_value cA (c t.sr)" by auto
from eigen_value_norm_sr[OF ev(2)] t.eigen_value_norm_sr[OF ev(1)]
show id: "t.sr = sr" by auto
from t.eigen_vector_z_sr[unfolded id, folded w_def] show "transpose A *v w = sr *s w"
unfolding eigen_vector_def by auto
from t.z_pos[folded w_def] show "lt_vec 0 w" by auto
qed

lemma c_cmod_id: "a ∈ ℝ ⟹ Re a ≥ 0 ⟹ c (cmod a) = a" by (auto simp: Reals_def)

lemma pos_rowvector_mult_0: assumes lt: "lt_vec 0 x"
and 0: "(rowvector x :: real ^ 'n ^ 'n) *v y = 0" (is "?x *v _ = 0") and le: "le_vec 0 y"
shows "y = 0"
proof -
{
fix i
assume "y \$ i ≠ 0"
with le have yi: "y \$ i > 0" by (auto simp: order.strict_iff_order)
have "0 = (?x *v y) \$ i" unfolding 0 by simp
also have "… = (∑j∈UNIV. x \$ j * y \$ j)"
unfolding rowvector_def matrix_vector_mult_def by simp
also have "… > 0"
by (rule sum_pos2[of _ i], insert yi lt le, auto intro!: mult_nonneg_nonneg
simp: order.strict_iff_order)
finally have False by simp
}
thus ?thesis by (auto simp: vec_eq_iff)
qed

lemma pos_matrix_mult_0: assumes le: "⋀ i j. B \$ i \$ j ≥ 0"
and lt: "lt_vec 0 x"
and 0: "B *v x = 0"
shows "B = 0"
proof -
{
fix i j
assume "B \$ i \$ j ≠ 0"
with le have gt: "B \$ i \$ j > 0" by (auto simp: order.strict_iff_order)
have "0 = (B *v x) \$ i" unfolding 0 by simp
also have "… = (∑j∈UNIV. B \$ i \$ j * x \$ j)"
unfolding matrix_vector_mult_def by simp
also have "… > 0"
by (rule sum_pos2[of _ j], insert gt lt le, auto intro!: mult_nonneg_nonneg
simp: order.strict_iff_order)
finally have False by simp
}
thus "B = 0" unfolding vec_eq_iff by auto
qed

lemma eigen_value_smaller_matrix: assumes B: "⋀ i j. 0 ≤ B \$ i \$ j ∧ B \$ i \$ j ≤ A \$ i \$ j"
and AB: "A ≠ B"
and ev: "eigen_value (map_matrix c B) sigma"
shows "cmod sigma < sr"
proof -
let ?B = "map_matrix c B"
define σ where "σ = ?sr"
have "real_non_neg_mat ?B" unfolding real_non_neg_mat_def elements_mat_h_def
by (auto simp: map_matrix_def map_vector_def B)
from perron_frobenius[OF this, folded σ_def] obtain x where ev_sr: "eigen_vector ?B x (c σ)"
and rnn: "real_non_neg_vec x" by auto
define y where "y = norm_v x"
from rnn have xy: "x = map_vector c y"
unfolding real_non_neg_vec_def vec_elements_h_def y_def
by (auto simp: map_vector_def vec_eq_iff c_cmod_id)
from spectral_radius_max[OF ev, folded σ_def] have sigma_sigma: "cmod sigma ≤ σ" .
from ev_sr[unfolded xy of_real_hom.eigen_vector_hom]
have ev_B: "eigen_vector B y σ" .
from ev_B[unfolded eigen_vector_def] have ev_B': "B *v y = σ *s y" by auto
have ypos: "y \$ i ≥ 0" for i unfolding y_def by (auto simp: map_vector_def)
from ev_B this have y: "y ∈ X" unfolding eigen_vector_def X_def by auto

have BA: "(B *v y) \$ i ≤ (A *v y) \$ i" for i
unfolding matrix_vector_mult_def vec_lambda_beta
by (rule sum_mono, rule mult_right_mono, insert B ypos, auto)
hence le_vec: "le_vec (σ *s y) (A *v y)" unfolding ev_B' by auto
from rho_le_x_Ax_imp_rho_le_rx[OF y le_vec]
have "σ ≤ r y" by auto
also have "… ≤ sr" using y by (rule sr_max)
finally have sig_le_sr: "σ ≤ sr" .
{
assume "σ = sr"
hence r_sr: "r y = sr" and sr_sig: "sr = σ" using ‹σ ≤ r y› ‹r y ≤ sr› by auto
from sr_u_pos[OF y r_sr] have pos: "lt_vec 0 y" .
from sr_imp_eigen_vector[OF y r_sr] have ev': "eigen_vector A y sr" .
have "(A - B) *v y = A *v y - B *v y" unfolding matrix_vector_mult_def
by (auto simp: vec_eq_iff field_simps sum_subtractf)
also have "A *v y = sr *s y" using ev'[unfolded eigen_vector_def] by auto
also have "B *v y = sr *s y" unfolding ev_B' sr_sig ..
finally have id: "(A - B) *v y = 0" by simp
from pos_matrix_mult_0[OF _ pos id] assms(1-2) have False by auto
}
with sig_le_sr sigma_sigma show ?thesis by argo
qed

lemma charpoly_erase_mat_sr: "0 < poly (charpoly (erase_mat A i i)) sr"
proof -
let ?A = "erase_mat A i i"
let ?pos = "poly (charpoly ?A) sr"
{
from A_nonzero_fixed_j[of i] obtain k where "A \$ k \$ i ≠ 0" by auto
assume "A = ?A"
hence "A \$ k \$ i = ?A \$ k \$ i" by simp
also have "?A \$ k \$ i = 0" by (auto simp: erase_mat_def)
also have "A \$ k \$ i ≠ 0" by fact
finally have False by simp
}
hence AA: "A ≠ ?A" by auto
have le: "0 ≤ ?A \$ i \$ j ∧ ?A \$ i \$ j ≤ A \$ i \$ j" for i j
by (auto simp: erase_mat_def nonneg)
note ev_small = eigen_value_smaller_matrix[OF le AA]
{
fix rho :: real
assume "eigen_value ?A rho"
hence ev: "eigen_value (map_matrix c ?A) (c rho)"
unfolding eigen_value_def using of_real_hom.eigen_vector_hom[of ?A _ rho] by auto
from ev_small[OF this] have "abs rho < sr" by auto
} note ev_small_real = this
have pos0: "?pos ≠ 0"
using ev_small_real[of sr] by (auto simp: eigen_value_root_charpoly)
{
define p where "p = charpoly ?A"
assume pos: "?pos < 0"
hence neg: "poly p sr < 0" unfolding p_def by auto
from degree_monic_charpoly[of ?A] have mon: "monic p" and deg: "degree p ≠ 0" unfolding p_def by auto
let ?f = "poly p"
have cont: "continuous_on {a..b} ?f" for a b by (auto intro: continuous_intros)
from pos have le: "?f sr ≤ 0" by (auto simp: p_def)
from mon have lc: "lead_coeff p > 0" by auto
from poly_pinfty_ge[OF this deg, of 0] obtain z where lez: "⋀ x. z ≤ x ⟹ 0 ≤ ?f x" by auto
define y where "y = max z sr"
have yr: "y ≥ sr" and "y ≥ z" unfolding y_def by auto
from lez[OF this(2)] have y0: "?f y ≥ 0" .
from IVT'[of ?f, OF le y0 yr cont] obtain x where ge: "x ≥ sr" and rt: "?f x = 0"
unfolding p_def by auto
hence "eigen_value ?A x" unfolding p_def by (simp add: eigen_value_root_charpoly)
from ev_small_real[OF this] ge have False by auto
}
with pos0 show ?thesis by argo
qed

lemma multiplicity_sr_1: "order sr (charpoly A) = 1"
proof -
{
assume "poly (pderiv (charpoly A)) sr = 0"
hence "0 = poly (monom 1 1 * pderiv (charpoly A)) sr" by simp
also have "… = sum (λ i. poly (charpoly (erase_mat A i i)) sr) UNIV"
unfolding pderiv_char_poly_erase_mat poly_sum ..
also have "… > 0"
by (rule sum_pos, (force simp: charpoly_erase_mat_sr)+)
finally have False by simp
}
hence nZ: "poly (pderiv (charpoly A)) sr ≠ 0" and nZ': "pderiv (charpoly A) ≠ 0" by auto
from eigen_vector_z_sr have "eigen_value A sr" unfolding eigen_value_def ..
from this[unfolded eigen_value_root_charpoly]
have "poly (charpoly A) sr = 0" .
hence "order sr (charpoly A) ≠ 0" unfolding order_root using nZ' by auto
from order_pderiv[OF nZ' this] order_0I[OF nZ]
show ?thesis by simp
qed

proof -
from eigen_vector_z_sr_c have "eigen_value cA (c sr)"
unfolding eigen_value_def by auto
have sr: "sr ≤ spectral_radius cA" by auto
show ?thesis by force
qed

lemma le_vec_A_mu: assumes y: "y ∈ X" and le: "le_vec (A *v y) (mu *s y)"
shows "sr ≤ mu" "lt_vec 0 y"
"mu = sr ∨ A *v y = mu *s y ⟹ mu = sr ∧ A *v y = mu *s y"
proof -
let ?w = "rowvector w"
let ?w' = "columnvector w"
have "?w ** A = transpose (transpose (?w ** A))"
unfolding transpose_transpose by simp
also have "transpose (?w ** A) = transpose A ** transpose ?w"
by (rule matrix_transpose_mul)
also have "transpose ?w = columnvector w" by (rule transpose_rowvector)
also have "transpose A ** … = columnvector (transpose A *v w)"
unfolding dot_rowvector_columnvector[symmetric] ..
also have "transpose A *v w = sr *s w" unfolding w by simp
also have "transpose (columnvector …) = rowvector (sr *s w)"
unfolding transpose_def columnvector_def rowvector_def vector_scalar_mult_def by auto
finally have 1: "?w ** A = rowvector (sr *s w)" .
have "sr *s (?w *v y) = ?w ** A *v y" unfolding 1
by (auto simp: rowvector_def vector_scalar_mult_def matrix_vector_mult_def vec_eq_iff
sum_distrib_left mult.assoc)
also have "… = ?w *v (A *v y)" by (simp add: matrix_vector_mul_assoc)
finally have eq1: "sr *s (rowvector w *v y) = rowvector w *v (A *v y)" .
have "le_vec (rowvector w *v (A *v y)) (?w *v (mu *s y))"
by (rule le_vec_mono_left[OF _ le], insert w(2), auto simp: rowvector_def order.strict_iff_order)
also have "?w *v (mu *s y) = mu *s (?w *v y)" by (simp add: algebra_simps vec.scale)
finally have le1: "le_vec (rowvector w *v (A *v y)) (mu *s (?w *v y))" .
from le1[unfolded eq1[symmetric]]
have 2: "le_vec (sr *s (?w *v y)) (mu *s (?w *v y))" .
{
from y obtain i where yi: "y \$ i > 0" and y: "⋀ j. y \$ j ≥ 0" unfolding X_def
by (auto simp: order.strict_iff_order vec_eq_iff)
from w(2) have wi: "w \$ i > 0" and w: "⋀ j. w \$ j ≥ 0"
by (auto simp: order.strict_iff_order)
have "(?w *v y) \$ i > 0" using yi y wi w
by (auto simp: matrix_vector_mult_def rowvector_def
intro!: sum_pos2[of _ i] mult_nonneg_nonneg)
moreover from 2[rule_format, of i] have "sr * (?w *v y) \$ i ≤ mu * (?w *v y) \$ i" by simp
ultimately have "sr ≤ mu" by simp
}
thus *: "sr ≤ mu" .
define cc where "cc = (mu + 1)^ N"
define n where "n = N"
from * sr_pos have mu: "mu ≥ 0" "mu > 0" by auto
hence cc: "cc > 0" unfolding cc_def by simp
from y have "pow_A_1 y ∈ pow_A_1 ` X" by auto
from Y_pos_main[OF this] have lt: "0 < (A1n *v y) \$ i" for i by (simp add: pow_A_1_def)
have le: "le_vec (A1n *v y) (cc *s y)" unfolding cc_def A1n_def n_def[symmetric]
proof (induct n)
case (Suc n)
let ?An = "matpow (A + mat 1) n"
let ?mu = "(mu + 1)"
have id': "matpow (A + mat 1) (Suc n) *v y = A *v (?An *v y) + ?An *v y" (is "?a = ?b + ?c")
matrix_vector_mul_assoc[symmetric])
have "le_vec ?b (?mu^n *s (A *v y))"
using le_vec_mono_left[OF nonneg Suc] by (simp add: algebra_simps vec.scale)
moreover have "le_vec (?mu^n *s (A *v y)) (?mu^n *s (mu *s y))"
using le mu by auto
moreover have id: "?mu^n *s (mu *s y) = (?mu^n * mu) *s y" by simp
from le_vec_trans[OF calculation[unfolded id]]
have le1: "le_vec ?b ((?mu^n * mu) *s y)" .
from Suc have le2: "le_vec ?c ((mu + 1) ^ n *s y)" .
have le: "le_vec ?a ((?mu^n * mu) *s y + ?mu^n *s y)"
unfolding id' using add_mono[OF le1[rule_format] le2[rule_format]] by auto
have id'': "(?mu^n * mu) *s y + ?mu^n *s y = ?mu^Suc n *s y" by (simp add: algebra_simps)
show ?case using le unfolding id'' .
have lt: "0 < cc * y \$ i" for i using lt[of i] le[rule_format, of i] by auto
have "y \$ i > 0" for i using lt[of i] cc by (rule zero_less_mult_pos)
thus "lt_vec 0 y" by auto
assume **: "mu = sr ∨ A *v y = mu *s y"
{
assume "A *v y = mu *s y"
with y have "eigen_vector A y mu" unfolding X_def eigen_vector_def by auto
hence "eigen_vector cA (map_vector c y) (c mu)" unfolding of_real_hom.eigen_vector_hom .
from eigen_vector_norm_sr[OF this] * have "mu = sr" by auto
}
with ** have mu_sr: "mu = sr" by auto
from eq1[folded vector_smult_distrib]
have 0: "?w *v (sr *s y - A *v y) = 0"
unfolding matrix_vector_right_distrib_diff by simp
have le0: "le_vec 0 (sr *s y - A *v y)" using assms(2)[unfolded mu_sr] by auto
have "sr *s y - A *v y = 0" using pos_rowvector_mult_0[OF w(2) 0 le0] .
hence ev_y: "A *v y = sr *s y" by auto
show "mu = sr ∧ A *v y = mu *s y" using ev_y mu_sr by auto
qed

lemma nonnegative_eigenvector_has_ev_sr: assumes "eigen_vector A v mu" and le: "le_vec 0 v"
shows "mu = sr"
proof -
from assms(1)[unfolded eigen_vector_def] have v: "v ≠ 0" and ev: "A *v v = mu *s v" by auto
from le v have v: "v ∈ X" unfolding X_def by auto
from ev have "le_vec (A *v v) (mu *s v)" by auto
from le_vec_A_mu[OF v this] ev show ?thesis by auto
qed

lemma similar_matrix_rotation: assumes ev: "eigen_value cA α" and α: "cmod α = sr"
shows "similar_matrix (cis (arg α) *k cA) cA"
proof -
from ev obtain y where ev: "eigen_vector cA y α" unfolding eigen_value_def by auto
let ?y = "norm_v y"
note maps = map_vector_def map_matrix_def
define yp where "yp = norm_v y"
let ?yp = "map_vector c yp"
have yp: "yp ∈ X" unfolding yp_def by (rule norm_v_X[OF ev])
from ev[unfolded eigen_vector_def] have ev_y: "cA *v y = α *s y" by auto
from ev_le_vec[OF ev, unfolded α, folded yp_def]
have 1: "le_vec (sr *s yp) (A *v yp)" by simp
from rho_le_x_Ax_imp_rho_le_rx[OF yp 1] have "sr ≤ r yp" by auto
with ev_inequalities[OF ev, folded yp_def]
have 2: "r yp = sr" by auto
have ev_yp: "A *v yp = sr *s yp"
and pos_yp: "lt_vec 0 yp"
using sr_imp_eigen_vector_main[OF yp 2] sr_u_pos[OF yp 2] by auto
define D where "D = diagvector (λ j. cis (arg (y \$ j)))"
define inv_D where "inv_D = diagvector (λ j. cis (- arg (y \$ j)))"
have DD: "inv_D ** D = mat 1" "D ** inv_D = mat 1" unfolding D_def inv_D_def
by (auto simp add: diagvector_eq_mat cis_mult)
{
fix i
have "(D *v ?yp) \$ i = cis (arg (y \$ i)) * c (cmod (y \$ i))"
unfolding D_def yp_def by (simp add: maps)
also have "… = y \$ i" by (simp add: cis_mult_cmod_id)
also note calculation
}
hence y_D_yp: "y = D *v ?yp" by (auto simp: vec_eq_iff)
define φ where "φ = arg α"
let ?φ = "cis (- φ)"
have [simp]: "cis (- φ) * rcis sr φ = sr" unfolding cis_rcis_eq rcis_mult by simp
have α: "α = rcis sr φ" unfolding φ_def α[symmetric] rcis_cmod_arg ..
define F where "F = ?φ *k (inv_D ** cA ** D)"
have "cA *v (D *v ?yp) = α *s y" unfolding y_D_yp[symmetric] ev_y by simp
also have "inv_D *v … = α *s ?yp"
unfolding vector_smult_distrib y_D_yp matrix_vector_mul_assoc DD matrix_vector_mul_lid ..
also have "?φ *s … = sr *s ?yp" unfolding α by simp
also have "… = map_vector c (sr *s yp)" unfolding vec_eq_iff by (auto simp: maps)
also have "… = cA *v ?yp" unfolding ev_yp[symmetric] by (auto simp: maps matrix_vector_mult_def)
finally have F: "F *v ?yp = cA *v ?yp" unfolding F_def matrix_scalar_vector_ac[symmetric]
unfolding matrix_vector_mul_assoc[symmetric] vector_smult_distrib .
have prod: "inv_D ** cA ** D = (χ i j. cis (- arg (y \$ i)) * cA \$ i \$ j * cis (arg (y \$ j)))"
unfolding inv_D_def D_def diagvector_mult_right diagvector_mult_left by simp
{
fix i j
have "cmod (F \$ i \$ j) = cmod (?φ * cA \$h i \$h j * (cis (- arg (y \$h i)) * cis (arg (y \$h j))))"
unfolding F_def prod vec_lambda_beta matrix_scalar_mult_def
by (simp only: ac_simps)
also have "… = A \$ i \$ j" unfolding cis_mult unfolding norm_mult by simp
also note calculation
}
hence FA: "map_matrix norm F = A" unfolding maps by auto
let ?F = "map_matrix c (map_matrix norm F)"
let ?G = "?F - F"
let ?Re = "map_matrix Re"
from F[folded FA] have 0: "?G *v ?yp = 0" unfolding matrix_diff_vect_distrib by simp
have "?Re ?G *v yp = map_vector Re (?G *v ?yp)"
unfolding maps matrix_vector_mult_def vec_lambda_beta Re_sum by auto
also have "… = 0" unfolding 0 by (simp add: vec_eq_iff maps)
finally have 0: "?Re ?G *v yp = 0" .
have "?Re ?G = 0"
by (rule pos_matrix_mult_0[OF _ pos_yp 0], auto simp: maps complex_Re_le_cmod)
hence "?F = F" by (auto simp: maps vec_eq_iff cmod_eq_Re)
with FA have AF: "cA = F" by simp
from arg_cong[OF this, of "λ A. cis φ *k A"]
have sim: "cis φ *k cA = inv_D ** cA ** D" unfolding F_def matrix.scale_scale cis_mult
by simp
have "similar_matrix (cis φ *k cA) cA" unfolding similar_matrix_def similar_matrix_wit_def
sim
by (rule exI[of _ inv_D], rule exI[of _ D], auto simp: DD)
thus ?thesis unfolding φ_def .
qed

lemma assumes ev: "eigen_value cA α" and α: "cmod α = sr"
shows maximal_eigen_value_order_1: "order α (charpoly cA) = 1"
and maximal_eigen_value_rotation: "eigen_value cA (x * cis (arg α)) = eigen_value cA x"
"eigen_value cA (x / cis (arg α)) = eigen_value cA x"
proof -
let ?a = "cis (arg α)"
let ?p = "charpoly cA"
from similar_matrix_rotation[OF ev α]
have "similar_matrix (?a *k cA) cA" .
from similar_matrix_charpoly[OF this]
have id: "charpoly (?a *k cA) = ?p" .
have a: "?a ≠ 0" by simp
from order_charpoly_smult[OF this, of _ cA, unfolded id]
have order_neg: "order x ?p = order (x / ?a) ?p" for x .
have order_pos: "order x ?p = order (x * ?a) ?p" for x
using order_neg[symmetric, of "x * ?a"] by simp
note order_neg[of α]
also have id: "α / ?a = sr" unfolding α[symmetric]
by (metis a cis_mult_cmod_id nonzero_mult_div_cancel_left)
also have sr: "order … ?p = 1" unfolding multiplicity_sr_1[symmetric] charpoly_of_real
by (rule map_poly_inj_idom_divide_hom.order_hom, unfold_locales)
finally show *: "order α ?p = 1" .
show "eigen_value cA (x * ?a) = eigen_value cA x" using order_pos
unfolding eigen_value_root_charpoly order_root by auto
show "eigen_value cA (x / ?a) = eigen_value cA x" using order_neg
unfolding eigen_value_root_charpoly order_root by auto
qed

lemma maximal_eigen_values_group: assumes M: "M = {ev :: complex. eigen_value cA ev ∧ cmod ev = sr}"
and a: "rcis sr α ∈ M"
and b: "rcis sr β ∈ M"
shows "rcis sr (α + β) ∈ M" "rcis sr (α - β) ∈ M" "rcis sr 0 ∈ M"
proof -
{
fix a
assume *: "rcis sr a ∈ M"
have id: "cis (arg (rcis sr a)) = cis a"
by (smt * M mem_Collect_eq nonzero_mult_div_cancel_left of_real_eq_0_iff
rcis_cmod_arg rcis_def sr_pos)
from *[unfolded assms] have "eigen_value cA (rcis sr a)" "cmod (rcis sr a) = sr" by auto
from maximal_eigen_value_rotation[OF this, unfolded id]
have "eigen_value cA (x * cis a) = eigen_value cA x"
"eigen_value cA (x / cis a) = eigen_value cA x" for x by auto
} note * = this
from *(1)[OF b, of "rcis sr α"] a show "rcis sr (α + β) ∈ M" unfolding M by auto
from *(2)[OF a, of "rcis sr α"] a show "rcis sr 0 ∈ M" unfolding M by auto
from *(2)[OF b, of "rcis sr α"] a show "rcis sr (α - β) ∈ M" unfolding M by auto
qed

lemma maximal_eigen_value_roots_of_unity_rotation:
assumes M: "M = {ev :: complex. eigen_value cA ev ∧ cmod ev = sr}"
and kM: "k = card M"
shows "k ≠ 0"
"k ≤ CARD('n)"
"∃ f. charpoly A = (monom 1 k - [:sr^k:]) * f
∧ (∀ x. poly (map_poly c f) x = 0 ⟶ cmod x < sr)"
"M = (*) (c sr) ` (λ i. (cis (of_nat i * 2 * pi / k))) ` {0 ..< k}"
"M = (*) (c sr) ` { x :: complex. x ^ k = 1}"
"(*) (cis (2 * pi / k)) ` Spectrum cA = Spectrum cA"
unfolding kM
proof -
let ?M = "card M"
note fin = finite_spectrum[of cA]
note char = degree_monic_charpoly[of cA]
have "?M ≤ card (Collect (eigen_value cA))"
by (rule card_mono[OF fin], unfold M, auto)
also have "Collect (eigen_value cA) = {x. poly (charpoly cA) x = 0}"
unfolding eigen_value_root_charpoly by auto
also have "card … ≤ degree (charpoly cA)"
by (rule poly_roots_degree, insert char, auto)
also have "… = CARD('n)" using char by simp
finally show "?M ≤ CARD ('n)" .
from finite_subset[OF _ fin, of M]
have finM: "finite M" unfolding M by blast
from finite_distinct_list[OF this]
obtain m where Mm: "M = set m" and dist: "distinct m" by auto
from Mm dist have card: "?M = length m" by (auto simp: distinct_card)
have sr: "sr ∈ set m" using eigen_value_sr_c sr_pos unfolding Mm[symmetric] M by auto
define s where "s = sort_key arg m"
define a where "a = map arg s"
let ?k = "length a"
from dist Mm card sr have s: "M = set s" "distinct s"  "sr ∈ set s"
and card: "?M = ?k"
and sorted: "sorted a"
unfolding s_def a_def by auto
have map_s: "map ((*) (c sr)) (map cis a) = s" unfolding map_map o_def a_def
proof (rule map_idI)
fix x
assume "x ∈ set s"
from this[folded s(1), unfolded M]
have id: "cmod x = sr" by auto
show "sr * cis (arg x) = x"
by (subst (5) rcis_cmod_arg[symmetric], unfold id[symmetric] rcis_def, simp)
qed
from s(2)[folded map_s, unfolded distinct_map] have a: "distinct a" "inj_on cis (set a)" by auto
from s(3) obtain aa a' where a_split: "a = aa # a'" unfolding a_def by (cases s, auto)
from arg_bounded have bounded: "x ∈ set a ⟹ - pi < x ∧ x ≤ pi" for x unfolding a_def by auto
from bounded[of aa, unfolded a_split] have aa: "- pi < aa ∧ aa ≤ pi" by auto
let ?aa = "aa + 2 * pi"
define args where "args = a @ [?aa]"
let ?diff = "λ i. args ! Suc i - args ! i"
have bnd: "x ∈ set a ⟹ x < ?aa" for x using aa bounded[of x] by auto
hence aa_a: "?aa ∉ set a" by fast
have sorted: "sorted args" unfolding args_def using sorted unfolding sorted_append
by (insert bnd, auto simp: order.strict_iff_order)
have dist: "distinct args" using a aa_a unfolding args_def distinct_append by auto
have sum: "(∑ i < ?k. ?diff i) = 2 * pi"
unfolding sum_lessThan_telescope args_def a_split by simp
have k: "?k ≠ 0" unfolding a_split by auto
let ?A = "?diff ` {..< ?k}"
let ?Min = "Min ?A"
define Min where "Min = ?Min"
have "?Min = (?k * ?Min) / ?k" using k by auto
also have "?k * ?Min = (∑ i < ?k. ?Min)" by auto
also have "… / ?k ≤ (∑ i < ?k. ?diff i) / ?k"
by (rule divide_right_mono[OF sum_mono[OF Min_le]], auto)
also have "… = 2 * pi / ?k" unfolding sum ..
finally have Min: "Min ≤ 2 * pi / ?k" unfolding Min_def by auto
have lt: "i < ?k ⟹ args ! i < args ! (Suc i)" for i
using sorted[unfolded sorted_iff_nth_mono, rule_format, of i "Suc i"]
dist[unfolded distinct_conv_nth, rule_format, of "Suc i" i] by (auto simp: args_def)
let ?c = "λ i. rcis sr (args ! i)"
have hda[simp]: "hd a = aa" unfolding a_split by simp
have Min0: "Min > 0" using lt unfolding Min_def by (subst Min_gr_iff, insert k, auto)
have Min_A: "Min ∈ ?A" unfolding Min_def by (rule Min_in, insert k, auto)
{
fix i :: nat
assume i: "i < length args"
hence "?c i = rcis sr ((a @ [hd a]) ! i)"
by (cases "i = ?k", auto simp: args_def nth_append rcis_def)
also have "… ∈ set (map (rcis sr) (a @ [hd a]))" using i
unfolding args_def set_map unfolding set_conv_nth by auto
also have "… = rcis sr ` set a" unfolding a_split by auto
also have "… = M" unfolding s(1) map_s[symmetric] set_map image_image
by (rule image_cong[OF refl], auto simp: rcis_def)
finally have "?c i ∈ M" by auto
} note ciM = this
{
fix i :: nat
assume i: "i < ?k"
hence "i < length args" "Suc i < length args" unfolding args_def by auto
from maximal_eigen_values_group[OF M ciM[OF this(2)] ciM[OF this(1)]]
have "rcis sr (?diff i) ∈ M" by simp
}
hence Min_M: "rcis sr Min ∈ M" using Min_A by force
have rcisM: "rcis sr (of_nat n * Min) ∈ M" for n
proof (induct n)
case 0
show ?case using sr Mm by auto
next
case (Suc n)
have *: "rcis sr (of_nat (Suc n) * Min) = rcis sr (of_nat n * Min) * cis Min"
from maximal_eigen_values_group(1)[OF M Suc Min_M]
show ?case unfolding * by simp
qed
let ?list = "map (rcis sr) (map (λ i. of_nat i * Min) [0 ..< ?k])"
define list where "list = ?list"
have len: "length ?list = ?M" unfolding card by simp
from sr_pos have sr0: "sr ≠ 0" by auto
{
fix i
assume i: "i < ?k"
hence *: "0 ≤ real i * Min" using Min0 by auto
from i have "real i < real ?k" by auto
from mult_strict_right_mono[OF this Min0]
have "real i * Min < real ?k * Min" by simp
also have "… ≤ real ?k * (2 * pi / real ?k)"
by (rule mult_left_mono[OF Min], auto)
also have "… = 2 * pi" using k by simp
finally have "real i * Min < 2 * pi" .
note * this
} note prod_pi = this
have dist: "distinct ?list"
unfolding distinct_map[of "rcis sr"]
proof (rule conjI[OF _ inj_on_subset[OF rcis_inj_on[OF sr0]]])
show "distinct (map (λ i. of_nat i * Min) [0 ..< ?k])" using Min0
by (auto simp: distinct_map inj_on_def)
show "set (map (λi. real i * Min) [0..<?k]) ⊆ {0..<2 * pi}" using prod_pi
by auto
qed
with len have card': "card (set ?list) = ?M" using distinct_card by fastforce
have listM: "set ?list ⊆ M" using rcisM by auto
from card_subset_eq[OF finM listM card']
have M_list: "M = set ?list" ..
let ?piM = "2 * pi / ?M"
{
assume "Min ≠ ?piM"
with Min have lt: "Min < 2 * pi / ?k" unfolding card by simp
from k have "0 < real ?k" by auto
from mult_strict_left_mono[OF lt this] k Min0
have k: "0 ≤ ?k * Min" "?k * Min < 2 * pi" by auto
from rcisM[of ?k, unfolded M_list] have "rcis sr (?k * Min) ∈ set ?list" by auto
then obtain i where i: "i < ?k" and id: "rcis sr (?k * Min) = rcis sr (i * Min)" by auto
from inj_onD[OF inj_on_subset[OF rcis_inj_on[OF sr0], of "{?k * Min, i * Min}"] id]
prod_pi[OF i] k
have "?k * Min = i * Min" by auto
with Min0 i have False by auto
}
hence Min: "Min = ?piM" by auto
show cM: "?M ≠ 0" unfolding card using k by auto
let ?f = "(λ i. cis (of_nat i * 2 * pi / ?M))"
note M_list
also have "set ?list = (*) (c sr) ` (λ i. cis (of_nat i * Min)) ` {0 ..< ?k}"
unfolding set_map image_image
by (rule image_cong, insert sr_pos, auto simp: rcis_mult rcis_def)
finally show M_cis: "M = (*) (c sr) ` ?f ` {0 ..< ?M}"
unfolding card Min by (simp add: mult.assoc)
thus M_pow: "M = (*) (c sr) ` { x :: complex. x ^ ?M = 1}" using roots_of_unity[OF cM] by simp
let ?rphi = "rcis sr (2 * pi / ?M)"
let ?phi = "cis (2 * pi / ?M)"
from Min_M[unfolded Min]
have ev: "eigen_value cA ?rphi" unfolding M by auto
have cm: "cmod ?rphi = sr" using sr_pos by simp
have id: "cis (arg ?rphi) = cis (arg ?phi) * cmod ?phi"
unfolding arg_rcis_cis[OF sr_pos] by simp
also have "… = ?phi" unfolding cis_mult_cmod_id ..
finally have id: "cis (arg ?rphi) = ?phi" .
define phi where "phi = ?phi"
have phi: "phi ≠ 0" unfolding phi_def by auto
note max = maximal_eigen_value_rotation[OF ev cm, unfolded id phi_def[symmetric]]
have "((*) phi) ` Spectrum cA = Spectrum cA" (is "?L = ?R")
proof -
{
fix x
have *: "x ∈ ?L ⟹ x ∈ ?R" for x using max(2)[of x] phi unfolding Spectrum_def by auto
moreover
{
assume "x ∈ ?R"
hence "eigen_value cA x" unfolding Spectrum_def by auto
from this[folded max(2)[of x]] have "x / phi ∈ ?R" unfolding Spectrum_def by auto
from imageI[OF this, of "(*) phi"]
have "x ∈ ?L" using phi by auto
}
note this *
}
thus ?thesis by blast
qed
from this[unfolded phi_def]
show "(*) (cis (2 * pi / real (card M))) ` Spectrum cA = Spectrum cA" .
let ?p = "monom 1 k - [:sr^k:]"
let ?cp = "monom 1 k - [:(c sr)^k:]"
let ?one = "1 :: complex"
let ?list = "map (rcis sr) (map (λ i. of_nat i * ?piM) [0 ..< card M])"
interpret c: field_hom c ..
interpret p: map_poly_inj_idom_divide_hom c ..
have cp: "?cp = map_poly c ?p" by (simp add: hom_distribs)
have M_list: "M = set ?list" using M_list[unfolded Min card[symmetric]] .
have dist: "distinct ?list" using dist[unfolded Min card[symmetric]] .
have k0: "k ≠ 0" using k[folded card] assms by auto
have "?cp = (monom 1 k + (- [:(c sr)^k:]))" by simp
also have "degree … = k"
by (subst degree_add_eq_left, insert k0, auto simp: degree_monom_eq)
finally have deg: "degree ?cp = k" .
from deg k0 have cp0: "?cp ≠ 0" by auto
have "{x. poly ?cp x = 0} = {x. x^k = (c sr)^k}" unfolding poly_diff poly_monom
by simp
also have "… ⊆ M"
proof -
{
fix x
assume id: "x^k = (c sr)^k"
from sr_pos k0 have "(c sr)^k ≠ 0" by auto
with arg_cong[OF id, of "λ x. x / (c sr)^k"]
have "(x / c sr)^k = 1"
unfolding power_divide by auto
hence "c sr * (x / c sr) ∈ M"
by (subst M_pow, unfold kM[symmetric], blast)
also have "c sr * (x / c sr) = x" using sr_pos by auto
finally have "x ∈ M" .
}
thus ?thesis by auto
qed
finally have cp_M: "{x. poly ?cp x = 0} ⊆ M" .
have "k = card (set ?list)" unfolding distinct_card[OF dist] by (simp add: kM)
also have "… ≤ card {x. poly ?cp x = 0}"
proof (rule card_mono[OF poly_roots_finite[OF cp0]])
{
fix x
assume "x ∈ set ?list"
then obtain i where x: "x = rcis sr (real i * ?piM)" by auto
have "x^k = (c sr)^k" unfolding x DeMoivre2 kM
by simp (metis mult.assoc of_real_power rcis_times_2pi)
hence "poly ?cp x = 0" unfolding poly_diff poly_monom by simp
}
thus "set ?list ⊆ {x. poly ?cp x = 0}" by auto
qed
finally have k_card: "k ≤ card {x. poly ?cp x = 0}" .
from k_card cp_M finM have M_id: "M = {x. poly ?cp x = 0}"
unfolding kM by (metis card_seteq)
have dvdc: "?cp dvd charpoly cA"
proof (rule poly_roots_dvd[OF cp0 deg k_card])
from cp_M
show "{x. poly ?cp x = 0} ⊆ {x. poly (charpoly cA) x = 0}"
unfolding M eigen_value_root_charpoly by auto
qed
from this[unfolded charpoly_of_real cp p.hom_dvd_iff]
have dvd: "?p dvd charpoly A" .
from this[unfolded dvd_def] obtain f where
decomp: "charpoly A = ?p * f" by blast
let ?f = "map_poly c f"
have decompc: "charpoly cA = ?cp * ?f" unfolding charpoly_of_real decomp p.hom_mult cp ..
show "∃ f. charpoly A = (monom 1 ?M - [:sr^?M:]) * f ∧ (∀ x. poly (map_poly c f) x = 0 ⟶ cmod x < sr)"
unfolding kM[symmetric]
proof (intro exI conjI allI impI, rule decomp)
fix x
assume f: "poly ?f x = 0"
hence ev: "eigen_value cA x"
unfolding decompc p.hom_mult eigen_value_root_charpoly by auto
hence le: "cmod x ≤ sr" using eigen_value_norm_sr by auto
{
assume max: "cmod x = sr"
hence "x ∈ M" unfolding M using ev by auto
hence "poly ?cp x = 0" unfolding M_id by auto
hence dvd1: "[: -x, 1 :] dvd ?cp" unfolding poly_eq_0_iff_dvd by auto
from f[unfolded poly_eq_0_iff_dvd]
have dvd2: "[: -x, 1 :] dvd ?f" by auto
from char have 0: "charpoly cA ≠ 0" by auto
from mult_dvd_mono[OF dvd1 dvd2] have "[: -x, 1 :]^2 dvd (charpoly cA)"
unfolding decompc power2_eq_square .
from order_max[OF this 0] maximal_eigen_value_order_1[OF ev max]
have False by auto
}
with le show "cmod x < sr" by argo
qed
qed

lemmas pf_main =
eigen_value_sr eigen_vector_z_sr (* sr is eigenvalue *)
eigen_value_norm_sr  (* it is maximal among all complex eigenvalues *)
z_pos    (* it's eigenvector is positive *)
multiplicity_sr_1 (* the algebr. multiplicity is 1 *)
nonnegative_eigenvector_has_ev_sr (* every non-negative real eigenvector has sr as eigenvalue *)
maximal_eigen_value_order_1 (* all maximal eigenvalues have order 1 *)
maximal_eigen_value_roots_of_unity_rotation
(* the maximal eigenvalues are precisely the k-th roots of unity for some k ≤ dim A *)