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" by (simp add: relpow_commute) 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) thus ?case using Suc by (simp add: matrix_add_ldistrib matrix_mul_rid) 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 by (simp add: compact_eq_bounded_closed) 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" by (induct n, auto simp: matrix_add_rdistrib matrix_add_ldistrib matrix_mul_rid matrix_mul_lid 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" by (simp add: comm_semiring_class.distrib) also have "… ≤ sr * ?y $ i + ?f i * ?y $ i" proof (rule add_left_mono[OF mult_right_mono]) 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" let ?sr = "spectral_radius ?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 lemma sr_spectral_radius: "sr = spectral_radius cA" proof - from eigen_vector_z_sr_c have "eigen_value cA (c sr)" unfolding eigen_value_def by auto from spectral_radius_max[OF this] have sr: "sr ≤ spectral_radius cA" by auto with spectral_radius_ev[of cA] eigen_vector_norm_sr 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") by (simp add: matrix_add_ldistrib matrix_mul_rid matrix_add_vect_distrib matpow_1_commute 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'' . qed (simp add: matrix_vector_mul_lid) 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" by (simp add: rcis_mult ring_distribs add.commute) 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 *) lemmas pf_main_connect = pf_main(1,3,5,7,8-10)[unfolded sr_spectral_radius] sr_pos[unfolded sr_spectral_radius] end end