Theory Perron_Frobenius_Aux
section ‹Perron-Frobenius Theorem›
subsection ‹Auxiliary Notions›
text ‹We define notions like non-negative real-valued matrix, both
in JNF and in HMA. These notions will be linked via HMA-connect.›
theory Perron_Frobenius_Aux
imports HMA_Connect
begin
definition real_nonneg_mat :: "complex mat ⇒ bool" where
"real_nonneg_mat A ≡ ∀ a ∈ elements_mat A. a ∈ ℝ ∧ Re a ≥ 0"
definition real_nonneg_vec :: "complex Matrix.vec ⇒ bool" where
"real_nonneg_vec v ≡ ∀ a ∈ vec_elements v. a ∈ ℝ ∧ Re a ≥ 0"
definition real_non_neg_vec :: "complex ^ 'n ⇒ bool" where
"real_non_neg_vec v ≡ (∀ a ∈ vec_elements_h v. a ∈ ℝ ∧ Re a ≥ 0)"
definition real_non_neg_mat :: "complex ^ 'nr ^ 'nc ⇒ bool" where
"real_non_neg_mat A ≡ (∀ a ∈ elements_mat_h A. a ∈ ℝ ∧ Re a ≥ 0)"
lemma real_non_neg_matD: assumes "real_non_neg_mat A"
shows "A $h i $h j ∈ ℝ" "Re (A $h i $h j) ≥ 0"
using assms unfolding real_non_neg_mat_def elements_mat_h_def by auto
definition nonneg_mat :: "'a :: linordered_idom mat ⇒ bool" where
"nonneg_mat A ≡ ∀ a ∈ elements_mat A. a ≥ 0"
definition non_neg_mat :: "'a :: linordered_idom ^ 'nr ^ 'nc ⇒ bool" where
"non_neg_mat A ≡ (∀ a ∈ elements_mat_h A. a ≥ 0)"
context includes lifting_syntax
begin
lemma HMA_real_non_neg_mat [transfer_rule]:
"((HMA_M :: complex mat ⇒ complex ^ 'nc ^ 'nr ⇒ bool) ===> (=))
real_nonneg_mat real_non_neg_mat"
unfolding real_nonneg_mat_def[abs_def] real_non_neg_mat_def[abs_def]
by transfer_prover
lemma HMA_real_non_neg_vec [transfer_rule]:
"((HMA_V :: complex Matrix.vec ⇒ complex ^ 'n ⇒ bool) ===> (=))
real_nonneg_vec real_non_neg_vec"
unfolding real_nonneg_vec_def[abs_def] real_non_neg_vec_def[abs_def]
by transfer_prover
lemma HMA_non_neg_mat [transfer_rule]:
"((HMA_M :: 'a :: linordered_idom mat ⇒ 'a ^ 'nc ^ 'nr ⇒ bool) ===> (=))
nonneg_mat non_neg_mat"
unfolding nonneg_mat_def[abs_def] non_neg_mat_def[abs_def]
by transfer_prover
end
primrec matpow :: "'a::semiring_1^'n^'n ⇒ nat ⇒ 'a^'n^'n" where
matpow_0: "matpow A 0 = mat 1" |
matpow_Suc: "matpow A (Suc n) = (matpow A n) ** A"
context includes lifting_syntax
begin
lemma HMA_pow_mat[transfer_rule]:
"((HMA_M ::'a::{semiring_1} mat ⇒ 'a^'n^'n ⇒ bool) ===> (=) ===> HMA_M) pow_mat matpow"
proof -
{
fix A :: "'a mat" and A' :: "'a^'n^'n" and n :: nat
assume [transfer_rule]: "HMA_M A A'"
hence [simp]: "dim_row A = CARD('n)" unfolding HMA_M_def by simp
have "HMA_M (pow_mat A n) (matpow A' n)"
proof (induct n)
case (Suc n)
note [transfer_rule] = this
show ?case by (simp, transfer_prover)
qed (simp, transfer_prover)
}
thus ?thesis by blast
qed
end
lemma trancl_image:
"(i,j) ∈ R⇧+ ⟹ (f i, f j) ∈ (map_prod f f ` R)⇧+"
proof (induct rule: trancl_induct)
case (step j k)
from step(2) have "(f j, f k) ∈ map_prod f f ` R" by auto
from step(3) this show ?case by auto
qed auto
lemma inj_trancl_image: assumes inj: "inj f"
shows "(f i, f j) ∈ (map_prod f f ` R)⇧+ = ((i,j) ∈ R⇧+)" (is "?l = ?r")
proof
assume ?r from trancl_image[OF this] show ?l .
next
assume ?l from trancl_image[OF this, of "the_inv f"]
show ?r unfolding image_image prod.map_comp o_def the_inv_f_f[OF inj] by auto
qed
lemma matrix_add_rdistrib: "((B + C) ** A) = (B ** A) + (C ** A)"
by (vector matrix_matrix_mult_def sum.distrib[symmetric] field_simps)
lemma norm_smult: "norm ((a :: real) *s x) = abs a * norm x"
unfolding norm_vec_def
by (metis norm_scaleR norm_vec_def scalar_mult_eq_scaleR)
lemma nonneg_mat_mult:
"nonneg_mat A ⟹ nonneg_mat B ⟹ A ∈ carrier_mat nr n
⟹ B ∈ carrier_mat n nc ⟹ nonneg_mat (A * B)"
unfolding nonneg_mat_def
by (auto simp: elements_mat_def scalar_prod_def intro!: sum_nonneg)
lemma nonneg_mat_power: assumes "A ∈ carrier_mat n n" "nonneg_mat A"
shows "nonneg_mat (A ^⇩m k)"
proof (induct k)
case 0
thus ?case by (auto simp: nonneg_mat_def)
next
case (Suc k)
from nonneg_mat_mult[OF this assms(2) _ assms(1), of n] assms(1)
show ?case by auto
qed
lemma nonneg_matD: assumes "nonneg_mat A"
and "i < dim_row A" and "j < dim_col A"
shows "A $$ (i,j) ≥ 0"
using assms unfolding nonneg_mat_def elements_mat by auto
lemma (in comm_ring_hom) similar_mat_wit_hom: assumes
"similar_mat_wit A B C D"
shows "similar_mat_wit (mat⇩h A) (mat⇩h B) (mat⇩h C) (mat⇩h D)"
proof -
obtain n where n: "n = dim_row A" by auto
note * = similar_mat_witD[OF n assms]
from * have [simp]: "dim_row C = n" by auto
note C = *(6) note D = *(7)
note id = mat_hom_mult[OF C D] mat_hom_mult[OF D C]
note ** = *(1-3)[THEN arg_cong[of _ _ "mat⇩h"], unfolded id]
note mult = mult_carrier_mat[of _ n n]
note hom_mult = mat_hom_mult[of _ n n _ n]
show ?thesis unfolding similar_mat_wit_def Let_def unfolding **(3) using **(1,2)
by (auto simp: n[symmetric] hom_mult simp: *(4-) mult)
qed
lemma (in comm_ring_hom) similar_mat_hom:
"similar_mat A B ⟹ similar_mat (mat⇩h A) (mat⇩h B)"
using similar_mat_wit_hom[of A B C D for C D]
by (smt similar_mat_def)
lemma det_dim_1: assumes A: "A ∈ carrier_mat n n"
and n: "n = 1"
shows "Determinant.det A = A $$ (0,0)"
by (subst laplace_expansion_column[OF A[unfolded n], of 0], insert A n,
auto simp: cofactor_def mat_delete_def)
lemma det_dim_2: assumes A: "A ∈ carrier_mat n n"
and n: "n = 2"
shows "Determinant.det A = A $$ (0,0) * A $$ (1,1) - A $$ (0,1) * A $$ (1,0)"
proof -
have set: "(∑i<(2 :: nat). f i) = f 0 + f 1" for f
by (subst sum.cong[of _ "{0,1}" f f], auto)
show ?thesis
apply (subst laplace_expansion_column[OF A[unfolded n], of 0], insert A n,
auto simp: cofactor_def mat_delete_def set)
apply (subst (1 2) det_dim_1, auto)
done
qed
lemma jordan_nf_root_char_poly: fixes A :: "'a :: {semiring_no_zero_divisors, idom} mat"
assumes "jordan_nf A n_as"
and "(m, lam) ∈ set n_as"
shows "poly (char_poly A) lam = 0"
proof -
from assms have m0: "m ≠ 0" unfolding jordan_nf_def by force
from split_list[OF assms(2)] obtain as bs where nas: "n_as = as @ (m, lam) # bs" by auto
show ?thesis using m0
unfolding jordan_nf_char_poly[OF assms(1)] nas poly_prod_list prod_list_zero_iff by (auto simp: o_def)
qed
lemma inverse_power_tendsto_zero:
"(λx. inverse ((of_nat x :: 'a :: real_normed_div_algebra) ^ Suc d)) ⇢ 0"
proof (rule filterlim_compose[OF tendsto_inverse_0],
intro filterlim_at_infinity[THEN iffD2, of 0] allI impI, goal_cases)
case (2 r)
let ?r = "nat (ceiling r) + 1"
show ?case
proof (intro eventually_sequentiallyI[of ?r], unfold norm_power norm_of_nat)
fix x
assume r: "?r ≤ x"
hence x1: "real x ≥ 1" by auto
have "r ≤ real ?r" by linarith
also have "… ≤ x" using r by auto
also have "… ≤ real x ^ Suc d" using x1 by simp
finally show "r ≤ real x ^ Suc d" .
qed
qed simp
lemma inverse_of_nat_tendsto_zero:
"(λx. inverse (of_nat x :: 'a :: real_normed_div_algebra)) ⇢ 0"
using inverse_power_tendsto_zero[of 0] by auto
lemma poly_times_exp_tendsto_zero: assumes b: "norm (b :: 'a :: real_normed_field) < 1"
shows "(λ x. of_nat x ^ k * b ^ x) ⇢ 0"
proof (cases "b = 0")
case False
define nla where "nla = norm b"
define s where "s = sqrt nla"
from b False have nla: "0 < nla" "nla < 1" unfolding nla_def by auto
hence s: "0 < s" "s < 1" unfolding s_def by auto
{
fix x
have "s^x * s^x = sqrt (nla ^ (2 * x))"
unfolding s_def power_add[symmetric]
unfolding real_sqrt_power[symmetric]
by (rule arg_cong[of _ _ "λ x. sqrt (nla ^ x)"], simp)
also have "… = nla^x" unfolding power_mult real_sqrt_power
using nla by simp
finally have "nla^x = s^x * s^x" by simp
} note nla_s = this
show "(λx. of_nat x ^ k * b ^ x) ⇢ 0"
proof (rule tendsto_norm_zero_cancel, unfold norm_mult norm_power norm_of_nat nla_def[symmetric] nla_s
mult.assoc[symmetric])
from poly_exp_constant_bound[OF s, of 1 k] obtain p where
p: "real x ^ k * s^x ≤ p" for x by (auto simp: ac_simps)
have "norm (real x ^ k * s ^ x) = real x ^ k * s^x" for x using s by auto
with p have p: "norm (real x ^ k * s ^ x) ≤ p" for x by auto
from s have s: "norm s < 1" by auto
show "(λx. real x ^ k * s ^ x * s ^ x) ⇢ 0"
by (rule lim_null_mult_left_bounded[OF _ LIMSEQ_power_zero[OF s], of _ p], insert p, auto)
qed
next
case True
show ?thesis unfolding True
by (subst tendsto_cong[of _ "λ x. 0"], rule eventually_sequentiallyI[of 1], auto)
qed
lemma (in linorder_topology) tendsto_Min: assumes I: "I ≠ {}" and fin: "finite I"
shows "(⋀i. i ∈ I ⟹ (f i ⤏ a i) F) ⟹ ((λx. Min ((λ i. f i x) ` I)) ⤏
(Min (a ` I) :: 'a)) F"
using fin I
proof (induct rule: finite_induct)
case (insert i I)
hence i: "(f i ⤏ a i) F" by auto
show ?case
proof (cases "I = {}")
case True
show ?thesis unfolding True using i by auto
next
case False
have *: "Min (a ` insert i I) = min (a i) (Min (a ` I))" using False insert(1) by auto
have **: "(λx. Min ((λi. f i x) ` insert i I)) = (λx. min (f i x) (Min ((λi. f i x) ` I)))"
using False insert(1) by auto
have IH: "((λx. Min ((λi. f i x) ` I)) ⤏ Min (a ` I)) F"
using insert(3)[OF insert(4) False] by auto
show ?thesis unfolding * **
by (auto intro!: tendsto_min i IH)
qed
qed simp
lemma tendsto_mat_mult [tendsto_intros]:
"(f ⤏ a) F ⟹ (g ⤏ b) F ⟹ ((λx. f x ** g x) ⤏ a ** b) F"
for f :: "'a ⇒ 'b :: {semiring_1, real_normed_algebra} ^ 'n1 ^ 'n2"
unfolding matrix_matrix_mult_def[abs_def] by (auto intro!: tendsto_intros)
lemma tendsto_matpower [tendsto_intros]: "(f ⤏ a) F ⟹ ((λx. matpow (f x) n) ⤏ matpow a n) F"
for f :: "'a ⇒ 'b :: {semiring_1, real_normed_algebra} ^ 'n ^ 'n"
by (induct n, simp_all add: tendsto_mat_mult)
lemma continuous_matpow: "continuous_on R (λ A :: 'a :: {semiring_1, real_normed_algebra_1} ^ 'n ^ 'n. matpow A n)"
unfolding continuous_on_def by (auto intro!: tendsto_intros)
lemma vector_smult_distrib: "(A *v ((a :: 'a :: comm_ring_1) *s x)) = a *s ((A *v x))"
unfolding matrix_vector_mult_def vector_scalar_mult_def
by (simp add: ac_simps sum_distrib_left)
instance real :: ordered_semiring_strict
by (intro_classes, auto)
lemma poly_tendsto_pinfty: fixes p :: "real poly"
assumes "lead_coeff p > 0" "degree p ≠ 0"
shows "poly p ⇢ ∞"
unfolding Lim_PInfty
proof
fix b
show "∃N. ∀n≥N. ereal b ≤ ereal (poly p (real n))"
unfolding ereal_less_eq using poly_pinfty_ge[OF assms, of b]
by (meson of_nat_le_iff order_trans real_arch_simple)
qed
lemma div_lt_nat: "(j :: nat) < x * y ⟹ j div x < y"
by (simp add: less_mult_imp_div_less mult.commute)
definition diagvector :: "('n ⇒ 'a :: semiring_0) ⇒ 'a ^ 'n ^ 'n" where
"diagvector x = (χ i. χ j. if i = j then x i else 0)"
lemma diagvector_mult_vector[simp]: "diagvector x *v y = (χ i. x i * y $ i)"
unfolding diagvector_def matrix_vector_mult_def vec_eq_iff vec_lambda_beta
proof (rule, goal_cases)
case (1 i)
show ?case by (subst sum.remove[of _ i], auto)
qed
lemma diagvector_mult_left: "diagvector x ** A = (χ i j. x i * A $ i $ j)" (is "?A = ?B")
unfolding vec_eq_iff
proof (intro allI)
fix i j
show "?A $h i $h j = ?B $h i $h j"
unfolding map_vector_def diagvector_def matrix_matrix_mult_def vec_lambda_beta
by (subst sum.remove[of _ i], auto)
qed
lemma diagvector_mult_right: "A ** diagvector x = (χ i j. A $ i $ j * x j)" (is "?A = ?B")
unfolding vec_eq_iff
proof (intro allI)
fix i j
show "?A $h i $h j = ?B $h i $h j"
unfolding map_vector_def diagvector_def matrix_matrix_mult_def vec_lambda_beta
by (subst sum.remove[of _ j], auto)
qed
lemma diagvector_mult[simp]: "diagvector x ** diagvector y = diagvector (λ i. x i * y i)"
unfolding diagvector_mult_left unfolding diagvector_def by (auto simp: vec_eq_iff)
lemma diagvector_const[simp]: "diagvector (λ x. k) = mat k"
unfolding diagvector_def mat_def by auto
lemma diagvector_eq_mat: "diagvector x = mat a ⟷ x = (λ x. a)"
unfolding diagvector_def mat_def by (auto simp: vec_eq_iff)
lemma cmod_eq_Re: assumes "cmod x = Re x"
shows "of_real (Re x) = x"
proof (cases "Im x = 0")
case False
hence "(cmod x)^2 ≠ (Re x)^2" unfolding norm_complex_def by simp
from this[unfolded assms] show ?thesis by auto
qed (cases x, auto simp: norm_complex_def complex_of_real_def)
hide_fact (open) Matrix.vec_eq_iff
no_notation
vec_index (infixl ‹$› 100)
lemma spectral_radius_ev:
"∃ ev v. eigen_vector A v ev ∧ norm ev = spectral_radius A"
proof -
from non_empty_spectrum[of A] finite_spectrum[of A] have
"spectral_radius A ∈ norm ` (Collect (eigen_value A))"
unfolding spectral_radius_ev_def by auto
thus ?thesis unfolding eigen_value_def[abs_def] by auto
qed
lemma spectral_radius_max: assumes "eigen_value A v"
shows "norm v ≤ spectral_radius A"
proof -
from assms have "norm v ∈ norm ` (Collect (eigen_value A))" by auto
from Max_ge[OF _ this, folded spectral_radius_ev_def]
finite_spectrum[of A] show ?thesis by auto
qed
text ‹For Perron-Frobenius it is useful to use the linear norm, and
not the Euclidean norm.›
definition norm1 :: "'a :: real_normed_field ^ 'n ⇒ real" where
"norm1 v = (∑i∈UNIV. norm (v $ i))"
lemma norm1_ge_0: "norm1 v ≥ 0" unfolding norm1_def
by (rule sum_nonneg, auto)
lemma norm1_0[simp]: "norm1 0 = 0" unfolding norm1_def by auto
lemma norm1_nonzero: assumes "v ≠ 0"
shows "norm1 v > 0"
proof -
from ‹v ≠ 0› obtain i where vi: "v $ i ≠ 0" unfolding vec_eq_iff
using Finite_Cartesian_Product.vec_eq_iff zero_index by force
have "sum (λ i. norm (v $ i)) (UNIV - {i}) ≥ 0"
by (rule sum_nonneg, auto)
moreover have "norm (v $ i) > 0" using vi by auto
ultimately
have "0 < norm (v $ i) + sum (λ i. norm (v $ i)) (UNIV - {i})" by arith
also have "… = norm1 v" unfolding norm1_def
by (simp add: sum.remove)
finally show "norm1 v > 0" .
qed
lemma norm1_0_iff[simp]: "(norm1 v = 0) = (v = 0)"
using norm1_0 norm1_nonzero by (cases "v = 0", force+)
lemma norm1_scaleR[simp]: "norm1 (r *⇩R v) = abs r * norm1 v" unfolding norm1_def sum_distrib_left
by (rule sum.cong, auto)
lemma abs_norm1[simp]: "abs (norm1 v) = norm1 v" using norm1_ge_0[of v] by arith
lemma normalize_eigen_vector: assumes "eigen_vector (A :: 'a :: real_normed_field ^ 'n ^ 'n) v ev"
shows "eigen_vector A ((1 / norm1 v) *⇩R v) ev" "norm1 ((1 / norm1 v) *⇩R v) = 1"
proof -
let ?v = "(1 / norm1 v) *⇩R v"
from assms[unfolded eigen_vector_def]
have nz: "v ≠ 0" and id: "A *v v = ev *s v" by auto
from nz have norm1: "norm1 v ≠ 0" by auto
thus "norm1 ?v = 1" by simp
from norm1 nz have nz: "?v ≠ 0" by auto
have "A *v ?v = (1 / norm1 v) *⇩R (A *v v)"
by (auto simp: vec_eq_iff matrix_vector_mult_def real_vector.scale_sum_right)
also have "A *v v = ev *s v" unfolding id ..
also have "(1 / norm1 v) *⇩R (ev *s v) = ev *s ?v"
by (auto simp: vec_eq_iff)
finally show "eigen_vector A ?v ev" using nz unfolding eigen_vector_def by auto
qed
lemma norm1_cont[simp]: "isCont norm1 v" unfolding norm1_def[abs_def] by auto
lemma norm1_ge_norm: "norm1 v ≥ norm v" unfolding norm1_def norm_vec_def
by (rule L2_set_le_sum, auto)
text ‹The following continuity lemmas have been proven with hints from Fabian Immler.›
lemma tendsto_matrix_vector_mult[tendsto_intros]:
"((*v) (A :: 'a :: real_normed_algebra_1 ^ 'n ^ 'k) ⤏ A *v v) (at v within S)"
unfolding matrix_vector_mult_def[abs_def]
by (auto intro!: tendsto_intros)
lemma tendsto_matrix_matrix_mult[tendsto_intros]:
"((**) (A :: 'a :: real_normed_algebra_1 ^ 'n ^ 'k) ⤏ A ** B) (at B within S)"
unfolding matrix_matrix_mult_def[abs_def]
by (auto intro!: tendsto_intros)
lemma matrix_vect_scaleR: "(A :: 'a :: real_normed_algebra_1 ^ 'n ^ 'k) *v (a *⇩R v) = a *⇩R (A *v v)"
unfolding vec_eq_iff
by (auto simp: matrix_vector_mult_def scaleR_vec_def scaleR_sum_right
intro!: sum.cong)
lemma (in inj_semiring_hom) map_vector_0: "(map_vector hom v = 0) = (v = 0)"
unfolding vec_eq_iff map_vector_def by auto
lemma (in inj_semiring_hom) map_vector_inj: "(map_vector hom v = map_vector hom w) = (v = w)"
unfolding vec_eq_iff map_vector_def by auto
lemma (in semiring_hom) matrix_vector_mult_hom:
"(map_matrix hom A) *v (map_vector hom v) = map_vector hom (A *v v)"
by (transfer fixing: hom, auto simp: mult_mat_vec_hom)
lemma (in semiring_hom) vector_smult_hom:
"hom x *s (map_vector hom v) = map_vector hom (x *s v)"
by (transfer fixing: hom, auto simp: vec_hom_smult)
lemma (in inj_comm_ring_hom) eigen_vector_hom:
"eigen_vector (map_matrix hom A) (map_vector hom v) (hom x) = eigen_vector A v x"
unfolding eigen_vector_def matrix_vector_mult_hom vector_smult_hom map_vector_0 map_vector_inj
by auto
end