(* Title: HOL/Analysis/Determinants.thy Author: Amine Chaieb, University of Cambridge *) section \Traces, Determinant of square matrices and some properties\ theory Determinants imports Cartesian_Euclidean_Space "~~/src/HOL/Library/Permutations" begin subsection \Trace\ definition trace :: "'a::semiring_1^'n^'n \ 'a" where "trace A = sum (\i. ((A$i)$i)) (UNIV::'n set)" lemma trace_0: "trace (mat 0) = 0" by (simp add: trace_def mat_def) lemma trace_I: "trace (mat 1 :: 'a::semiring_1^'n^'n) = of_nat(CARD('n))" by (simp add: trace_def mat_def) lemma trace_add: "trace ((A::'a::comm_semiring_1^'n^'n) + B) = trace A + trace B" by (simp add: trace_def sum.distrib) lemma trace_sub: "trace ((A::'a::comm_ring_1^'n^'n) - B) = trace A - trace B" by (simp add: trace_def sum_subtractf) lemma trace_mul_sym: "trace ((A::'a::comm_semiring_1^'n^'m) ** B) = trace (B**A)" apply (simp add: trace_def matrix_matrix_mult_def) apply (subst sum.commute) apply (simp add: mult.commute) done text \Definition of determinant.\ definition det:: "'a::comm_ring_1^'n^'n \ 'a" where "det A = sum (\p. of_int (sign p) * setprod (\i. A$i$p i) (UNIV :: 'n set)) {p. p permutes (UNIV :: 'n set)}" text \A few general lemmas we need below.\ lemma setprod_permute: assumes p: "p permutes S" shows "setprod f S = setprod (f \ p) S" using assms by (fact setprod.permute) lemma setproduct_permute_nat_interval: fixes m n :: nat shows "p permutes {m..n} \ setprod f {m..n} = setprod (f \ p) {m..n}" by (blast intro!: setprod_permute) text \Basic determinant properties.\ lemma det_transpose: "det (transpose A) = det (A::'a::comm_ring_1 ^'n^'n)" proof - let ?di = "\A i j. A$i$j" let ?U = "(UNIV :: 'n set)" have fU: "finite ?U" by simp { fix p assume p: "p \ {p. p permutes ?U}" from p have pU: "p permutes ?U" by blast have sth: "sign (inv p) = sign p" by (metis sign_inverse fU p mem_Collect_eq permutation_permutes) from permutes_inj[OF pU] have pi: "inj_on p ?U" by (blast intro: subset_inj_on) from permutes_image[OF pU] have "setprod (\i. ?di (transpose A) i (inv p i)) ?U = setprod (\i. ?di (transpose A) i (inv p i)) (p ` ?U)" by simp also have "\ = setprod ((\i. ?di (transpose A) i (inv p i)) \ p) ?U" unfolding setprod.reindex[OF pi] .. also have "\ = setprod (\i. ?di A i (p i)) ?U" proof - { fix i assume i: "i \ ?U" from i permutes_inv_o[OF pU] permutes_in_image[OF pU] have "((\i. ?di (transpose A) i (inv p i)) \ p) i = ?di A i (p i)" unfolding transpose_def by (simp add: fun_eq_iff) } then show "setprod ((\i. ?di (transpose A) i (inv p i)) \ p) ?U = setprod (\i. ?di A i (p i)) ?U" by (auto intro: setprod.cong) qed finally have "of_int (sign (inv p)) * (setprod (\i. ?di (transpose A) i (inv p i)) ?U) = of_int (sign p) * (setprod (\i. ?di A i (p i)) ?U)" using sth by simp } then show ?thesis unfolding det_def apply (subst sum_permutations_inverse) apply (rule sum.cong) apply (rule refl) apply blast done qed lemma det_lowerdiagonal: fixes A :: "'a::comm_ring_1^('n::{finite,wellorder})^('n::{finite,wellorder})" assumes ld: "\i j. i < j \ A$i$j = 0" shows "det A = setprod (\i. A$i$i) (UNIV:: 'n set)" proof - let ?U = "UNIV:: 'n set" let ?PU = "{p. p permutes ?U}" let ?pp = "\p. of_int (sign p) * setprod (\i. A$i$p i) (UNIV :: 'n set)" have fU: "finite ?U" by simp from finite_permutations[OF fU] have fPU: "finite ?PU" . have id0: "{id} \ ?PU" by (auto simp add: permutes_id) { fix p assume p: "p \ ?PU - {id}" from p have pU: "p permutes ?U" and pid: "p \ id" by blast+ from permutes_natset_le[OF pU] pid obtain i where i: "p i > i" by (metis not_le) from ld[OF i] have ex:"\i \ ?U. A$i$p i = 0" by blast from setprod_zero[OF fU ex] have "?pp p = 0" by simp } then have p0: "\p \ ?PU - {id}. ?pp p = 0" by blast from sum.mono_neutral_cong_left[OF fPU id0 p0] show ?thesis unfolding det_def by (simp add: sign_id) qed lemma det_upperdiagonal: fixes A :: "'a::comm_ring_1^'n::{finite,wellorder}^'n::{finite,wellorder}" assumes ld: "\i j. i > j \ A$i$j = 0" shows "det A = setprod (\i. A$i$i) (UNIV:: 'n set)" proof - let ?U = "UNIV:: 'n set" let ?PU = "{p. p permutes ?U}" let ?pp = "(\p. of_int (sign p) * setprod (\i. A$i$p i) (UNIV :: 'n set))" have fU: "finite ?U" by simp from finite_permutations[OF fU] have fPU: "finite ?PU" . have id0: "{id} \ ?PU" by (auto simp add: permutes_id) { fix p assume p: "p \ ?PU - {id}" from p have pU: "p permutes ?U" and pid: "p \ id" by blast+ from permutes_natset_ge[OF pU] pid obtain i where i: "p i < i" by (metis not_le) from ld[OF i] have ex:"\i \ ?U. A$i$p i = 0" by blast from setprod_zero[OF fU ex] have "?pp p = 0" by simp } then have p0: "\p \ ?PU -{id}. ?pp p = 0" by blast from sum.mono_neutral_cong_left[OF fPU id0 p0] show ?thesis unfolding det_def by (simp add: sign_id) qed lemma det_diagonal: fixes A :: "'a::comm_ring_1^'n^'n" assumes ld: "\i j. i \ j \ A$i$j = 0" shows "det A = setprod (\i. A$i$i) (UNIV::'n set)" proof - let ?U = "UNIV:: 'n set" let ?PU = "{p. p permutes ?U}" let ?pp = "\p. of_int (sign p) * setprod (\i. A$i$p i) (UNIV :: 'n set)" have fU: "finite ?U" by simp from finite_permutations[OF fU] have fPU: "finite ?PU" . have id0: "{id} \ ?PU" by (auto simp add: permutes_id) { fix p assume p: "p \ ?PU - {id}" then have "p \ id" by simp then obtain i where i: "p i \ i" unfolding fun_eq_iff by auto from ld [OF i [symmetric]] have ex:"\i \ ?U. A$i$p i = 0" by blast from setprod_zero [OF fU ex] have "?pp p = 0" by simp } then have p0: "\p \ ?PU - {id}. ?pp p = 0" by blast from sum.mono_neutral_cong_left[OF fPU id0 p0] show ?thesis unfolding det_def by (simp add: sign_id) qed lemma det_I: "det (mat 1 :: 'a::comm_ring_1^'n^'n) = 1" proof - let ?A = "mat 1 :: 'a::comm_ring_1^'n^'n" let ?U = "UNIV :: 'n set" let ?f = "\i j. ?A$i$j" { fix i assume i: "i \ ?U" have "?f i i = 1" using i by (vector mat_def) } then have th: "setprod (\i. ?f i i) ?U = setprod (\x. 1) ?U" by (auto intro: setprod.cong) { fix i j assume i: "i \ ?U" and j: "j \ ?U" and ij: "i \ j" have "?f i j = 0" using i j ij by (vector mat_def) } then have "det ?A = setprod (\i. ?f i i) ?U" using det_diagonal by blast also have "\ = 1" unfolding th setprod.neutral_const .. finally show ?thesis . qed lemma det_0: "det (mat 0 :: 'a::comm_ring_1^'n^'n) = 0" by (simp add: det_def setprod_zero) lemma det_permute_rows: fixes A :: "'a::comm_ring_1^'n^'n" assumes p: "p permutes (UNIV :: 'n::finite set)" shows "det (\ i. A$p i :: 'a^'n^'n) = of_int (sign p) * det A" apply (simp add: det_def sum_distrib_left mult.assoc[symmetric]) apply (subst sum_permutations_compose_right[OF p]) proof (rule sum.cong) let ?U = "UNIV :: 'n set" let ?PU = "{p. p permutes ?U}" fix q assume qPU: "q \ ?PU" have fU: "finite ?U" by simp from qPU have q: "q permutes ?U" by blast from p q have pp: "permutation p" and qp: "permutation q" by (metis fU permutation_permutes)+ from permutes_inv[OF p] have ip: "inv p permutes ?U" . have "setprod (\i. A$p i$ (q \ p) i) ?U = setprod ((\i. A$p i$(q \ p) i) \ inv p) ?U" by (simp only: setprod_permute[OF ip, symmetric]) also have "\ = setprod (\i. A $ (p \ inv p) i $ (q \ (p \ inv p)) i) ?U" by (simp only: o_def) also have "\ = setprod (\i. A$i$q i) ?U" by (simp only: o_def permutes_inverses[OF p]) finally have thp: "setprod (\i. A$p i$ (q \ p) i) ?U = setprod (\i. A$i$q i) ?U" by blast show "of_int (sign (q \ p)) * setprod (\i. A$ p i$ (q \ p) i) ?U = of_int (sign p) * of_int (sign q) * setprod (\i. A$i$q i) ?U" by (simp only: thp sign_compose[OF qp pp] mult.commute of_int_mult) qed rule lemma det_permute_columns: fixes A :: "'a::comm_ring_1^'n^'n" assumes p: "p permutes (UNIV :: 'n set)" shows "det(\ i j. A$i$ p j :: 'a^'n^'n) = of_int (sign p) * det A" proof - let ?Ap = "\ i j. A$i$ p j :: 'a^'n^'n" let ?At = "transpose A" have "of_int (sign p) * det A = det (transpose (\ i. transpose A $ p i))" unfolding det_permute_rows[OF p, of ?At] det_transpose .. moreover have "?Ap = transpose (\ i. transpose A $ p i)" by (simp add: transpose_def vec_eq_iff) ultimately show ?thesis by simp qed lemma det_identical_rows: fixes A :: "'a::linordered_idom^'n^'n" assumes ij: "i \ j" and r: "row i A = row j A" shows "det A = 0" proof- have tha: "\(a::'a) b. a = b \ b = - a \ a = 0" by simp have th1: "of_int (-1) = - 1" by simp let ?p = "Fun.swap i j id" let ?A = "\ i. A $ ?p i" from r have "A = ?A" by (simp add: vec_eq_iff row_def Fun.swap_def) then have "det A = det ?A" by simp moreover have "det A = - det ?A" by (simp add: det_permute_rows[OF permutes_swap_id] sign_swap_id ij th1) ultimately show "det A = 0" by (metis tha) qed lemma det_identical_columns: fixes A :: "'a::linordered_idom^'n^'n" assumes ij: "i \ j" and r: "column i A = column j A" shows "det A = 0" apply (subst det_transpose[symmetric]) apply (rule det_identical_rows[OF ij]) apply (metis row_transpose r) done lemma det_zero_row: fixes A :: "'a::{idom, ring_char_0}^'n^'n" assumes r: "row i A = 0" shows "det A = 0" using r apply (simp add: row_def det_def vec_eq_iff) apply (rule sum.neutral) apply (auto simp: sign_nz) done lemma det_zero_column: fixes A :: "'a::{idom,ring_char_0}^'n^'n" assumes r: "column i A = 0" shows "det A = 0" apply (subst det_transpose[symmetric]) apply (rule det_zero_row [of i]) apply (metis row_transpose r) done lemma det_row_add: fixes a b c :: "'n::finite \ _ ^ 'n" shows "det((\ i. if i = k then a i + b i else c i)::'a::comm_ring_1^'n^'n) = det((\ i. if i = k then a i else c i)::'a::comm_ring_1^'n^'n) + det((\ i. if i = k then b i else c i)::'a::comm_ring_1^'n^'n)" unfolding det_def vec_lambda_beta sum.distrib[symmetric] proof (rule sum.cong) let ?U = "UNIV :: 'n set" let ?pU = "{p. p permutes ?U}" let ?f = "(\i. if i = k then a i + b i else c i)::'n \ 'a::comm_ring_1^'n" let ?g = "(\ i. if i = k then a i else c i)::'n \ 'a::comm_ring_1^'n" let ?h = "(\ i. if i = k then b i else c i)::'n \ 'a::comm_ring_1^'n" fix p assume p: "p \ ?pU" let ?Uk = "?U - {k}" from p have pU: "p permutes ?U" by blast have kU: "?U = insert k ?Uk" by blast { fix j assume j: "j \ ?Uk" from j have "?f j $ p j = ?g j $ p j" and "?f j $ p j= ?h j $ p j" by simp_all } then have th1: "setprod (\i. ?f i $ p i) ?Uk = setprod (\i. ?g i $ p i) ?Uk" and th2: "setprod (\i. ?f i $ p i) ?Uk = setprod (\i. ?h i $ p i) ?Uk" apply - apply (rule setprod.cong, simp_all)+ done have th3: "finite ?Uk" "k \ ?Uk" by auto have "setprod (\i. ?f i $ p i) ?U = setprod (\i. ?f i $ p i) (insert k ?Uk)" unfolding kU[symmetric] .. also have "\ = ?f k $ p k * setprod (\i. ?f i $ p i) ?Uk" apply (rule setprod.insert) apply simp apply blast done also have "\ = (a k $ p k * setprod (\i. ?f i $ p i) ?Uk) + (b k$ p k * setprod (\i. ?f i $ p i) ?Uk)" by (simp add: field_simps) also have "\ = (a k $ p k * setprod (\i. ?g i $ p i) ?Uk) + (b k$ p k * setprod (\i. ?h i $ p i) ?Uk)" by (metis th1 th2) also have "\ = setprod (\i. ?g i $ p i) (insert k ?Uk) + setprod (\i. ?h i $ p i) (insert k ?Uk)" unfolding setprod.insert[OF th3] by simp finally have "setprod (\i. ?f i $ p i) ?U = setprod (\i. ?g i $ p i) ?U + setprod (\i. ?h i $ p i) ?U" unfolding kU[symmetric] . then show "of_int (sign p) * setprod (\i. ?f i $ p i) ?U = of_int (sign p) * setprod (\i. ?g i $ p i) ?U + of_int (sign p) * setprod (\i. ?h i $ p i) ?U" by (simp add: field_simps) qed rule lemma det_row_mul: fixes a b :: "'n::finite \ _ ^ 'n" shows "det((\ i. if i = k then c *s a i else b i)::'a::comm_ring_1^'n^'n) = c * det((\ i. if i = k then a i else b i)::'a::comm_ring_1^'n^'n)" unfolding det_def vec_lambda_beta sum_distrib_left proof (rule sum.cong) let ?U = "UNIV :: 'n set" let ?pU = "{p. p permutes ?U}" let ?f = "(\i. if i = k then c*s a i else b i)::'n \ 'a::comm_ring_1^'n" let ?g = "(\ i. if i = k then a i else b i)::'n \ 'a::comm_ring_1^'n" fix p assume p: "p \ ?pU" let ?Uk = "?U - {k}" from p have pU: "p permutes ?U" by blast have kU: "?U = insert k ?Uk" by blast { fix j assume j: "j \ ?Uk" from j have "?f j $ p j = ?g j $ p j" by simp } then have th1: "setprod (\i. ?f i $ p i) ?Uk = setprod (\i. ?g i $ p i) ?Uk" apply - apply (rule setprod.cong) apply simp_all done have th3: "finite ?Uk" "k \ ?Uk" by auto have "setprod (\i. ?f i $ p i) ?U = setprod (\i. ?f i $ p i) (insert k ?Uk)" unfolding kU[symmetric] .. also have "\ = ?f k $ p k * setprod (\i. ?f i $ p i) ?Uk" apply (rule setprod.insert) apply simp apply blast done also have "\ = (c*s a k) $ p k * setprod (\i. ?f i $ p i) ?Uk" by (simp add: field_simps) also have "\ = c* (a k $ p k * setprod (\i. ?g i $ p i) ?Uk)" unfolding th1 by (simp add: ac_simps) also have "\ = c* (setprod (\i. ?g i $ p i) (insert k ?Uk))" unfolding setprod.insert[OF th3] by simp finally have "setprod (\i. ?f i $ p i) ?U = c* (setprod (\i. ?g i $ p i) ?U)" unfolding kU[symmetric] . then show "of_int (sign p) * setprod (\i. ?f i $ p i) ?U = c * (of_int (sign p) * setprod (\i. ?g i $ p i) ?U)" by (simp add: field_simps) qed rule lemma det_row_0: fixes b :: "'n::finite \ _ ^ 'n" shows "det((\ i. if i = k then 0 else b i)::'a::comm_ring_1^'n^'n) = 0" using det_row_mul[of k 0 "\i. 1" b] apply simp apply (simp only: vector_smult_lzero) done lemma det_row_operation: fixes A :: "'a::linordered_idom^'n^'n" assumes ij: "i \ j" shows "det (\ k. if k = i then row i A + c *s row j A else row k A) = det A" proof - let ?Z = "(\ k. if k = i then row j A else row k A) :: 'a ^'n^'n" have th: "row i ?Z = row j ?Z" by (vector row_def) have th2: "((\ k. if k = i then row i A else row k A) :: 'a^'n^'n) = A" by (vector row_def) show ?thesis unfolding det_row_add [of i] det_row_mul[of i] det_identical_rows[OF ij th] th2 by simp qed lemma det_row_span: fixes A :: "real^'n^'n" assumes x: "x \ span {row j A |j. j \ i}" shows "det (\ k. if k = i then row i A + x else row k A) = det A" proof - let ?U = "UNIV :: 'n set" let ?S = "{row j A |j. j \ i}" let ?d = "\x. det (\ k. if k = i then x else row k A)" let ?P = "\x. ?d (row i A + x) = det A" { fix k have "(if k = i then row i A + 0 else row k A) = row k A" by simp } then have P0: "?P 0" apply - apply (rule cong[of det, OF refl]) apply (vector row_def) done moreover { fix c z y assume zS: "z \ ?S" and Py: "?P y" from zS obtain j where j: "z = row j A" "i \ j" by blast let ?w = "row i A + y" have th0: "row i A + (c*s z + y) = ?w + c*s z" by vector have thz: "?d z = 0" apply (rule det_identical_rows[OF j(2)]) using j apply (vector row_def) done have "?d (row i A + (c*s z + y)) = ?d (?w + c*s z)" unfolding th0 .. then have "?P (c*s z + y)" unfolding thz Py det_row_mul[of i] det_row_add[of i] by simp } ultimately show ?thesis apply - apply (rule span_induct_alt[of ?P ?S, OF P0, folded scalar_mult_eq_scaleR]) apply blast apply (rule x) done qed text \ May as well do this, though it's a bit unsatisfactory since it ignores exact duplicates by considering the rows/columns as a set. \ lemma det_dependent_rows: fixes A:: "real^'n^'n" assumes d: "dependent (rows A)" shows "det A = 0" proof - let ?U = "UNIV :: 'n set" from d obtain i where i: "row i A \ span (rows A - {row i A})" unfolding dependent_def rows_def by blast { fix j k assume jk: "j \ k" and c: "row j A = row k A" from det_identical_rows[OF jk c] have ?thesis . } moreover { assume H: "\ i j. i \ j \ row i A \ row j A" have th0: "- row i A \ span {row j A|j. j \ i}" apply (rule span_neg) apply (rule set_rev_mp) apply (rule i) apply (rule span_mono) using H i apply (auto simp add: rows_def) done from det_row_span[OF th0] have "det A = det (\ k. if k = i then 0 *s 1 else row k A)" unfolding right_minus vector_smult_lzero .. with det_row_mul[of i "0::real" "\i. 1"] have "det A = 0" by simp } ultimately show ?thesis by blast qed lemma det_dependent_columns: assumes d: "dependent (columns (A::real^'n^'n))" shows "det A = 0" by (metis d det_dependent_rows rows_transpose det_transpose) text \Multilinearity and the multiplication formula.\ lemma Cart_lambda_cong: "(\x. f x = g x) \ (vec_lambda f::'a^'n) = (vec_lambda g :: 'a^'n)" by (rule iffD1[OF vec_lambda_unique]) vector lemma det_linear_row_sum: assumes fS: "finite S" shows "det ((\ i. if i = k then sum (a i) S else c i)::'a::comm_ring_1^'n^'n) = sum (\j. det ((\ i. if i = k then a i j else c i)::'a^'n^'n)) S" proof (induct rule: finite_induct[OF fS]) case 1 then show ?case apply simp unfolding sum.empty det_row_0[of k] apply rule done next case (2 x F) then show ?case by (simp add: det_row_add cong del: if_weak_cong) qed lemma finite_bounded_functions: assumes fS: "finite S" shows "finite {f. (\i \ {1.. (k::nat)}. f i \ S) \ (\i. i \ {1 .. k} \ f i = i)}" proof (induct k) case 0 have th: "{f. \i. f i = i} = {id}" by auto show ?case by (auto simp add: th) next case (Suc k) let ?f = "\(y::nat,g) i. if i = Suc k then y else g i" let ?S = "?f ` (S \ {f. (\i\{1..k}. f i \ S) \ (\i. i \ {1..k} \ f i = i)})" have "?S = {f. (\i\{1.. Suc k}. f i \ S) \ (\i. i \ {1.. Suc k} \ f i = i)}" apply (auto simp add: image_iff) apply (rule_tac x="x (Suc k)" in bexI) apply (rule_tac x = "\i. if i = Suc k then i else x i" in exI) apply auto done with finite_imageI[OF finite_cartesian_product[OF fS Suc.hyps(1)], of ?f] show ?case by metis qed lemma det_linear_rows_sum_lemma: assumes fS: "finite S" and fT: "finite T" shows "det ((\ i. if i \ T then sum (a i) S else c i):: 'a::comm_ring_1^'n^'n) = sum (\f. det((\ i. if i \ T then a i (f i) else c i)::'a^'n^'n)) {f. (\i \ T. f i \ S) \ (\i. i \ T \ f i = i)}" using fT proof (induct T arbitrary: a c set: finite) case empty have th0: "\x y. (\ i. if i \ {} then x i else y i) = (\ i. y i)" by vector from empty.prems show ?case unfolding th0 by (simp add: eq_id_iff) next case (insert z T a c) let ?F = "\T. {f. (\i \ T. f i \ S) \ (\i. i \ T \ f i = i)}" let ?h = "\(y,g) i. if i = z then y else g i" let ?k = "\h. (h(z),(\i. if i = z then i else h i))" let ?s = "\ k a c f. det((\ i. if i \ T then a i (f i) else c i)::'a^'n^'n)" let ?c = "\j i. if i = z then a i j else c i" have thif: "\a b c d. (if a \ b then c else d) = (if a then c else if b then c else d)" by simp have thif2: "\a b c d e. (if a then b else if c then d else e) = (if c then (if a then b else d) else (if a then b else e))" by simp from \z \ T\ have nz: "\i. i \ T \ i = z \ False" by auto have "det (\ i. if i \ insert z T then sum (a i) S else c i) = det (\ i. if i = z then sum (a i) S else if i \ T then sum (a i) S else c i)" unfolding insert_iff thif .. also have "\ = (\j\S. det (\ i. if i \ T then sum (a i) S else if i = z then a i j else c i))" unfolding det_linear_row_sum[OF fS] apply (subst thif2) using nz apply (simp cong del: if_weak_cong cong add: if_cong) done finally have tha: "det (\ i. if i \ insert z T then sum (a i) S else c i) = (\(j, f)\S \ ?F T. det (\ i. if i \ T then a i (f i) else if i = z then a i j else c i))" unfolding insert.hyps unfolding sum.cartesian_product by blast show ?case unfolding tha using \z \ T\ by (intro sum.reindex_bij_witness[where i="?k" and j="?h"]) (auto intro!: cong[OF refl[of det]] simp: vec_eq_iff) qed lemma det_linear_rows_sum: fixes S :: "'n::finite set" assumes fS: "finite S" shows "det (\ i. sum (a i) S) = sum (\f. det (\ i. a i (f i) :: 'a::comm_ring_1 ^ 'n^'n)) {f. \i. f i \ S}" proof - have th0: "\x y. ((\ i. if i \ (UNIV:: 'n set) then x i else y i) :: 'a^'n^'n) = (\ i. x i)" by vector from det_linear_rows_sum_lemma[OF fS, of "UNIV :: 'n set" a, unfolded th0, OF finite] show ?thesis by simp qed lemma matrix_mul_sum_alt: fixes A B :: "'a::comm_ring_1^'n^'n" shows "A ** B = (\ i. sum (\k. A$i$k *s B $ k) (UNIV :: 'n set))" by (vector matrix_matrix_mult_def sum_component) lemma det_rows_mul: "det((\ i. c i *s a i)::'a::comm_ring_1^'n^'n) = setprod (\i. c i) (UNIV:: 'n set) * det((\ i. a i)::'a^'n^'n)" proof (simp add: det_def sum_distrib_left cong add: setprod.cong, rule sum.cong) let ?U = "UNIV :: 'n set" let ?PU = "{p. p permutes ?U}" fix p assume pU: "p \ ?PU" let ?s = "of_int (sign p)" from pU have p: "p permutes ?U" by blast have "setprod (\i. c i * a i $ p i) ?U = setprod c ?U * setprod (\i. a i $ p i) ?U" unfolding setprod.distrib .. then show "?s * (\xa\?U. c xa * a xa $ p xa) = setprod c ?U * (?s* (\xa\?U. a xa $ p xa))" by (simp add: field_simps) qed rule lemma det_mul: fixes A B :: "'a::linordered_idom^'n^'n" shows "det (A ** B) = det A * det B" proof - let ?U = "UNIV :: 'n set" let ?F = "{f. (\i\ ?U. f i \ ?U) \ (\i. i \ ?U \ f i = i)}" let ?PU = "{p. p permutes ?U}" have fU: "finite ?U" by simp have fF: "finite ?F" by (rule finite) { fix p assume p: "p permutes ?U" have "p \ ?F" unfolding mem_Collect_eq permutes_in_image[OF p] using p[unfolded permutes_def] by simp } then have PUF: "?PU \ ?F" by blast { fix f assume fPU: "f \ ?F - ?PU" have fUU: "f ` ?U \ ?U" using fPU by auto from fPU have f: "\i \ ?U. f i \ ?U" "\i. i \ ?U \ f i = i" "\(\y. \!x. f x = y)" unfolding permutes_def by auto let ?A = "(\ i. A$i$f i *s B$f i) :: 'a^'n^'n" let ?B = "(\ i. B$f i) :: 'a^'n^'n" { assume fni: "\ inj_on f ?U" then obtain i j where ij: "f i = f j" "i \ j" unfolding inj_on_def by blast from ij have rth: "row i ?B = row j ?B" by (vector row_def) from det_identical_rows[OF ij(2) rth] have "det (\ i. A$i$f i *s B$f i) = 0" unfolding det_rows_mul by simp } moreover { assume fi: "inj_on f ?U" from f fi have fith: "\i j. f i = f j \ i = j" unfolding inj_on_def by metis note fs = fi[unfolded surjective_iff_injective_gen[OF fU fU refl fUU, symmetric]] { fix y from fs f have "\x. f x = y" by blast then obtain x where x: "f x = y" by blast { fix z assume z: "f z = y" from fith x z have "z = x" by metis } with x have "\!x. f x = y" by blast } with f(3) have "det (\ i. A$i$f i *s B$f i) = 0" by blast } ultimately have "det (\ i. A$i$f i *s B$f i) = 0" by blast } then have zth: "\ f\ ?F - ?PU. det (\ i. A$i$f i *s B$f i) = 0" by simp { fix p assume pU: "p \ ?PU" from pU have p: "p permutes ?U" by blast let ?s = "\p. of_int (sign p)" let ?f = "\q. ?s p * (\i\ ?U. A $ i $ p i) * (?s q * (\i\ ?U. B $ i $ q i))" have "(sum (\q. ?s q * (\i\ ?U. (\ i. A $ i $ p i *s B $ p i :: 'a^'n^'n) $ i $ q i)) ?PU) = (sum (\q. ?s p * (\i\ ?U. A $ i $ p i) * (?s q * (\i\ ?U. B $ i $ q i))) ?PU)" unfolding sum_permutations_compose_right[OF permutes_inv[OF p], of ?f] proof (rule sum.cong) fix q assume qU: "q \ ?PU" then have q: "q permutes ?U" by blast from p q have pp: "permutation p" and pq: "permutation q" unfolding permutation_permutes by auto have th00: "of_int (sign p) * of_int (sign p) = (1::'a)" "\a. of_int (sign p) * (of_int (sign p) * a) = a" unfolding mult.assoc[symmetric] unfolding of_int_mult[symmetric] by (simp_all add: sign_idempotent) have ths: "?s q = ?s p * ?s (q \ inv p)" using pp pq permutation_inverse[OF pp] sign_inverse[OF pp] by (simp add: th00 ac_simps sign_idempotent sign_compose) have th001: "setprod (\i. B$i$ q (inv p i)) ?U = setprod ((\i. B$i$ q (inv p i)) \ p) ?U" by (rule setprod_permute[OF p]) have thp: "setprod (\i. (\ i. A$i$p i *s B$p i :: 'a^'n^'n) $i $ q i) ?U = setprod (\i. A$i$p i) ?U * setprod (\i. B$i$ q (inv p i)) ?U" unfolding th001 setprod.distrib[symmetric] o_def permutes_inverses[OF p] apply (rule setprod.cong[OF refl]) using permutes_in_image[OF q] apply vector done show "?s q * setprod (\i. (((\ i. A$i$p i *s B$p i) :: 'a^'n^'n)$i$q i)) ?U = ?s p * (setprod (\i. A$i$p i) ?U) * (?s (q \ inv p) * setprod (\i. B$i$(q \ inv p) i) ?U)" using ths thp pp pq permutation_inverse[OF pp] sign_inverse[OF pp] by (simp add: sign_nz th00 field_simps sign_idempotent sign_compose) qed rule } then have th2: "sum (\f. det (\ i. A$i$f i *s B$f i)) ?PU = det A * det B" unfolding det_def sum_product by (rule sum.cong [OF refl]) have "det (A**B) = sum (\f. det (\ i. A $ i $ f i *s B $ f i)) ?F" unfolding matrix_mul_sum_alt det_linear_rows_sum[OF fU] by simp also have "\ = sum (\f. det (\ i. A$i$f i *s B$f i)) ?PU" using sum.mono_neutral_cong_left[OF fF PUF zth, symmetric] unfolding det_rows_mul by auto finally show ?thesis unfolding th2 . qed text \Relation to invertibility.\ lemma invertible_left_inverse: fixes A :: "real^'n^'n" shows "invertible A \ (\(B::real^'n^'n). B** A = mat 1)" by (metis invertible_def matrix_left_right_inverse) lemma invertible_righ_inverse: fixes A :: "real^'n^'n" shows "invertible A \ (\(B::real^'n^'n). A** B = mat 1)" by (metis invertible_def matrix_left_right_inverse) lemma invertible_det_nz: fixes A::"real ^'n^'n" shows "invertible A \ det A \ 0" proof - { assume "invertible A" then obtain B :: "real ^'n^'n" where B: "A ** B = mat 1" unfolding invertible_righ_inverse by blast then have "det (A ** B) = det (mat 1 :: real ^'n^'n)" by simp then have "det A \ 0" by (simp add: det_mul det_I) algebra } moreover { assume H: "\ invertible A" let ?U = "UNIV :: 'n set" have fU: "finite ?U" by simp from H obtain c i where c: "sum (\i. c i *s row i A) ?U = 0" and iU: "i \ ?U" and ci: "c i \ 0" unfolding invertible_righ_inverse unfolding matrix_right_invertible_independent_rows by blast have *: "\(a::real^'n) b. a + b = 0 \ -a = b" apply (drule_tac f="op + (- a)" in cong[OF refl]) apply (simp only: ab_left_minus add.assoc[symmetric]) apply simp done from c ci have thr0: "- row i A = sum (\j. (1/ c i) *s (c j *s row j A)) (?U - {i})" unfolding sum.remove[OF fU iU] sum_cmul apply - apply (rule vector_mul_lcancel_imp[OF ci]) apply (auto simp add: field_simps) unfolding * apply rule done have thr: "- row i A \ span {row j A| j. j \ i}" unfolding thr0 apply (rule span_sum) apply simp apply (rule span_mul [where 'a="real^'n", folded scalar_mult_eq_scaleR])+ apply (rule span_superset) apply auto done let ?B = "(\ k. if k = i then 0 else row k A) :: real ^'n^'n" have thrb: "row i ?B = 0" using iU by (vector row_def) have "det A = 0" unfolding det_row_span[OF thr, symmetric] right_minus unfolding det_zero_row[OF thrb] .. } ultimately show ?thesis by blast qed text \Cramer's rule.\ lemma cramer_lemma_transpose: fixes A:: "real^'n^'n" and x :: "real^'n" shows "det ((\ i. if i = k then sum (\i. x$i *s row i A) (UNIV::'n set) else row i A)::real^'n^'n) = x$k * det A" (is "?lhs = ?rhs") proof - let ?U = "UNIV :: 'n set" let ?Uk = "?U - {k}" have U: "?U = insert k ?Uk" by blast have fUk: "finite ?Uk" by simp have kUk: "k \ ?Uk" by simp have th00: "\k s. x$k *s row k A + s = (x$k - 1) *s row k A + row k A + s" by (vector field_simps) have th001: "\f k . (\x. if x = k then f k else f x) = f" by auto have "(\ i. row i A) = A" by (vector row_def) then have thd1: "det (\ i. row i A) = det A" by simp have thd0: "det (\ i. if i = k then row k A + (\i \ ?Uk. x $ i *s row i A) else row i A) = det A" apply (rule det_row_span) apply (rule span_sum) apply (rule span_mul [where 'a="real^'n", folded scalar_mult_eq_scaleR])+ apply (rule span_superset) apply auto done show "?lhs = x$k * det A" apply (subst U) unfolding sum.insert[OF fUk kUk] apply (subst th00) unfolding add.assoc apply (subst det_row_add) unfolding thd0 unfolding det_row_mul unfolding th001[of k "\i. row i A"] unfolding thd1 apply (simp add: field_simps) done qed lemma cramer_lemma: fixes A :: "real^'n^'n" shows "det((\ i j. if j = k then (A *v x)$i else A$i$j):: real^'n^'n) = x$k * det A" proof - let ?U = "UNIV :: 'n set" have *: "\c. sum (\i. c i *s row i (transpose A)) ?U = sum (\i. c i *s column i A) ?U" by (auto simp add: row_transpose intro: sum.cong) show ?thesis unfolding matrix_mult_vsum unfolding cramer_lemma_transpose[of k x "transpose A", unfolded det_transpose, symmetric] unfolding *[of "\i. x$i"] apply (subst det_transpose[symmetric]) apply (rule cong[OF refl[of det]]) apply (vector transpose_def column_def row_def) done qed lemma cramer: fixes A ::"real^'n^'n" assumes d0: "det A \ 0" shows "A *v x = b \ x = (\ k. det(\ i j. if j=k then b$i else A$i$j) / det A)" proof - from d0 obtain B where B: "A ** B = mat 1" "B ** A = mat 1" unfolding invertible_det_nz[symmetric] invertible_def by blast have "(A ** B) *v b = b" by (simp add: B matrix_vector_mul_lid) then have "A *v (B *v b) = b" by (simp add: matrix_vector_mul_assoc) then have xe: "\x. A *v x = b" by blast { fix x assume x: "A *v x = b" have "x = (\ k. det(\ i j. if j=k then b$i else A$i$j) / det A)" unfolding x[symmetric] using d0 by (simp add: vec_eq_iff cramer_lemma field_simps) } with xe show ?thesis by auto qed text \Orthogonality of a transformation and matrix.\ definition "orthogonal_transformation f \ linear f \ (\v w. f v \ f w = v \ w)" lemma orthogonal_transformation: "orthogonal_transformation f \ linear f \ (\(v::real ^_). norm (f v) = norm v)" unfolding orthogonal_transformation_def apply auto apply (erule_tac x=v in allE)+ apply (simp add: norm_eq_sqrt_inner) apply (simp add: dot_norm linear_add[symmetric]) done definition "orthogonal_matrix (Q::'a::semiring_1^'n^'n) \ transpose Q ** Q = mat 1 \ Q ** transpose Q = mat 1" lemma orthogonal_matrix: "orthogonal_matrix (Q:: real ^'n^'n) \ transpose Q ** Q = mat 1" by (metis matrix_left_right_inverse orthogonal_matrix_def) lemma orthogonal_matrix_id: "orthogonal_matrix (mat 1 :: _^'n^'n)" by (simp add: orthogonal_matrix_def transpose_mat matrix_mul_lid) lemma orthogonal_matrix_mul: fixes A :: "real ^'n^'n" assumes oA : "orthogonal_matrix A" and oB: "orthogonal_matrix B" shows "orthogonal_matrix(A ** B)" using oA oB unfolding orthogonal_matrix matrix_transpose_mul apply (subst matrix_mul_assoc) apply (subst matrix_mul_assoc[symmetric]) apply (simp add: matrix_mul_rid) done lemma orthogonal_transformation_matrix: fixes f:: "real^'n \ real^'n" shows "orthogonal_transformation f \ linear f \ orthogonal_matrix(matrix f)" (is "?lhs \ ?rhs") proof - let ?mf = "matrix f" let ?ot = "orthogonal_transformation f" let ?U = "UNIV :: 'n set" have fU: "finite ?U" by simp let ?m1 = "mat 1 :: real ^'n^'n" { assume ot: ?ot from ot have lf: "linear f" and fd: "\v w. f v \ f w = v \ w" unfolding orthogonal_transformation_def orthogonal_matrix by blast+ { fix i j let ?A = "transpose ?mf ** ?mf" have th0: "\b (x::'a::comm_ring_1). (if b then 1 else 0)*x = (if b then x else 0)" "\b (x::'a::comm_ring_1). x*(if b then 1 else 0) = (if b then x else 0)" by simp_all from fd[rule_format, of "axis i 1" "axis j 1", simplified matrix_works[OF lf, symmetric] dot_matrix_vector_mul] have "?A$i$j = ?m1 $ i $ j" by (simp add: inner_vec_def matrix_matrix_mult_def columnvector_def rowvector_def th0 sum.delta[OF fU] mat_def axis_def) } then have "orthogonal_matrix ?mf" unfolding orthogonal_matrix by vector with lf have ?rhs by blast } moreover { assume lf: "linear f" and om: "orthogonal_matrix ?mf" from lf om have ?lhs apply (simp only: orthogonal_matrix_def norm_eq orthogonal_transformation) apply (simp only: matrix_works[OF lf, symmetric]) apply (subst dot_matrix_vector_mul) apply (simp add: dot_matrix_product matrix_mul_lid) done } ultimately show ?thesis by blast qed lemma det_orthogonal_matrix: fixes Q:: "'a::linordered_idom^'n^'n" assumes oQ: "orthogonal_matrix Q" shows "det Q = 1 \ det Q = - 1" proof - have th: "\x::'a. x = 1 \ x = - 1 \ x*x = 1" (is "\x::'a. ?ths x") proof - fix x:: 'a have th0: "x * x - 1 = (x - 1) * (x + 1)" by (simp add: field_simps) have th1: "\(x::'a) y. x = - y \ x + y = 0" apply (subst eq_iff_diff_eq_0) apply simp done have "x * x = 1 \ x * x - 1 = 0" by simp also have "\ \ x = 1 \ x = - 1" unfolding th0 th1 by simp finally show "?ths x" .. qed from oQ have "Q ** transpose Q = mat 1" by (metis orthogonal_matrix_def) then have "det (Q ** transpose Q) = det (mat 1:: 'a^'n^'n)" by simp then have "det Q * det Q = 1" by (simp add: det_mul det_I det_transpose) then show ?thesis unfolding th . qed text \Linearity of scaling, and hence isometry, that preserves origin.\ lemma scaling_linear: fixes f :: "real ^'n \ real ^'n" assumes f0: "f 0 = 0" and fd: "\x y. dist (f x) (f y) = c * dist x y" shows "linear f" proof - { fix v w { fix x note fd[rule_format, of x 0, unfolded dist_norm f0 diff_0_right] } note th0 = this have "f v \ f w = c\<^sup>2 * (v \ w)" unfolding dot_norm_neg dist_norm[symmetric] unfolding th0 fd[rule_format] by (simp add: power2_eq_square field_simps)} note fc = this show ?thesis unfolding linear_iff vector_eq[where 'a="real^'n"] scalar_mult_eq_scaleR by (simp add: inner_add fc field_simps) qed lemma isometry_linear: "f (0:: real^'n) = (0:: real^'n) \ \x y. dist(f x) (f y) = dist x y \ linear f" by (rule scaling_linear[where c=1]) simp_all text \Hence another formulation of orthogonal transformation.\ lemma orthogonal_transformation_isometry: "orthogonal_transformation f \ f(0::real^'n) = (0::real^'n) \ (\x y. dist(f x) (f y) = dist x y)" unfolding orthogonal_transformation apply (rule iffI) apply clarify apply (clarsimp simp add: linear_0 linear_diff[symmetric] dist_norm) apply (rule conjI) apply (rule isometry_linear) apply simp apply simp apply clarify apply (erule_tac x=v in allE) apply (erule_tac x=0 in allE) apply (simp add: dist_norm) done text \Can extend an isometry from unit sphere.\ lemma isometry_sphere_extend: fixes f:: "real ^'n \ real ^'n" assumes f1: "\x. norm x = 1 \ norm (f x) = 1" and fd1: "\ x y. norm x = 1 \ norm y = 1 \ dist (f x) (f y) = dist x y" shows "\g. orthogonal_transformation g \ (\x. norm x = 1 \ g x = f x)" proof - { fix x y x' y' x0 y0 x0' y0' :: "real ^'n" assume H: "x = norm x *\<^sub>R x0" "y = norm y *\<^sub>R y0" "x' = norm x *\<^sub>R x0'" "y' = norm y *\<^sub>R y0'" "norm x0 = 1" "norm x0' = 1" "norm y0 = 1" "norm y0' = 1" "norm(x0' - y0') = norm(x0 - y0)" then have *: "x0 \ y0 = x0' \ y0' + y0' \ x0' - y0 \ x0 " by (simp add: norm_eq norm_eq_1 inner_add inner_diff) have "norm(x' - y') = norm(x - y)" apply (subst H(1)) apply (subst H(2)) apply (subst H(3)) apply (subst H(4)) using H(5-9) apply (simp add: norm_eq norm_eq_1) apply (simp add: inner_diff scalar_mult_eq_scaleR) unfolding * apply (simp add: field_simps) done } note th0 = this let ?g = "\x. if x = 0 then 0 else norm x *\<^sub>R f (inverse (norm x) *\<^sub>R x)" { fix x:: "real ^'n" assume nx: "norm x = 1" have "?g x = f x" using nx by auto } then have thfg: "\x. norm x = 1 \ ?g x = f x" by blast have g0: "?g 0 = 0" by simp { fix x y :: "real ^'n" { assume "x = 0" "y = 0" then have "dist (?g x) (?g y) = dist x y" by simp } moreover { assume "x = 0" "y \ 0" then have "dist (?g x) (?g y) = dist x y" apply (simp add: dist_norm) apply (rule f1[rule_format]) apply (simp add: field_simps) done } moreover { assume "x \ 0" "y = 0" then have "dist (?g x) (?g y) = dist x y" apply (simp add: dist_norm) apply (rule f1[rule_format]) apply (simp add: field_simps) done } moreover { assume z: "x \ 0" "y \ 0" have th00: "x = norm x *\<^sub>R (inverse (norm x) *\<^sub>R x)" "y = norm y *\<^sub>R (inverse (norm y) *\<^sub>R y)" "norm x *\<^sub>R f ((inverse (norm x) *\<^sub>R x)) = norm x *\<^sub>R f (inverse (norm x) *\<^sub>R x)" "norm y *\<^sub>R f (inverse (norm y) *\<^sub>R y) = norm y *\<^sub>R f (inverse (norm y) *\<^sub>R y)" "norm (inverse (norm x) *\<^sub>R x) = 1" "norm (f (inverse (norm x) *\<^sub>R x)) = 1" "norm (inverse (norm y) *\<^sub>R y) = 1" "norm (f (inverse (norm y) *\<^sub>R y)) = 1" "norm (f (inverse (norm x) *\<^sub>R x) - f (inverse (norm y) *\<^sub>R y)) = norm (inverse (norm x) *\<^sub>R x - inverse (norm y) *\<^sub>R y)" using z by (auto simp add: field_simps intro: f1[rule_format] fd1[rule_format, unfolded dist_norm]) from z th0[OF th00] have "dist (?g x) (?g y) = dist x y" by (simp add: dist_norm) } ultimately have "dist (?g x) (?g y) = dist x y" by blast } note thd = this show ?thesis apply (rule exI[where x= ?g]) unfolding orthogonal_transformation_isometry using g0 thfg thd apply metis done qed text \Rotation, reflection, rotoinversion.\ definition "rotation_matrix Q \ orthogonal_matrix Q \ det Q = 1" definition "rotoinversion_matrix Q \ orthogonal_matrix Q \ det Q = - 1" lemma orthogonal_rotation_or_rotoinversion: fixes Q :: "'a::linordered_idom^'n^'n" shows " orthogonal_matrix Q \ rotation_matrix Q \ rotoinversion_matrix Q" by (metis rotoinversion_matrix_def rotation_matrix_def det_orthogonal_matrix) text \Explicit formulas for low dimensions.\ lemma setprod_neutral_const: "setprod f {(1::nat)..1} = f 1" by simp lemma setprod_2: "setprod f {(1::nat)..2} = f 1 * f 2" by (simp add: eval_nat_numeral atLeastAtMostSuc_conv mult.commute) lemma setprod_3: "setprod f {(1::nat)..3} = f 1 * f 2 * f 3" by (simp add: eval_nat_numeral atLeastAtMostSuc_conv mult.commute) lemma det_1: "det (A::'a::comm_ring_1^1^1) = A$1$1" by (simp add: det_def of_nat_Suc sign_id) lemma det_2: "det (A::'a::comm_ring_1^2^2) = A$1$1 * A$2$2 - A$1$2 * A$2$1" proof - have f12: "finite {2::2}" "1 \ {2::2}" by auto show ?thesis unfolding det_def UNIV_2 unfolding sum_over_permutations_insert[OF f12] unfolding permutes_sing by (simp add: sign_swap_id sign_id swap_id_eq) qed lemma det_3: "det (A::'a::comm_ring_1^3^3) = A$1$1 * A$2$2 * A$3$3 + A$1$2 * A$2$3 * A$3$1 + A$1$3 * A$2$1 * A$3$2 - A$1$1 * A$2$3 * A$3$2 - A$1$2 * A$2$1 * A$3$3 - A$1$3 * A$2$2 * A$3$1" proof - have f123: "finite {2::3, 3}" "1 \ {2::3, 3}" by auto have f23: "finite {3::3}" "2 \ {3::3}" by auto show ?thesis unfolding det_def UNIV_3 unfolding sum_over_permutations_insert[OF f123] unfolding sum_over_permutations_insert[OF f23] unfolding permutes_sing by (simp add: sign_swap_id permutation_swap_id sign_compose sign_id swap_id_eq) qed end