fact stringlengths 8 1.54k | type stringclasses 19
values | library stringclasses 8
values | imports listlengths 1 10 | filename stringclasses 98
values | symbolic_name stringlengths 1 42 | docstring stringclasses 1
value |
|---|---|---|---|---|---|---|
center_mxPm n A (R : 'A_(m, n)) :
reflect (A \in R /\ forall B, B \in R -> B *m A = A *m B)
(A \in 'Z(R))%MS.
Proof.
rewrite sub_capmx; case R_A: (A \in R); last by right; case.
by apply: (iffP cent_mxP) => [cAR | [_ cAR]].
Qed.
Arguments center_mxP {m n A R}. | Lemma | algebra | [
"From HB Require Import structures",
"From mathcomp Require Import ssreflect ssrbool ssrfun eqtype ssrnat seq choice",
"From mathcomp Require Import fintype finfun bigop finset fingroup perm order",
"From mathcomp Require Import div prime binomial ssralg finalg zmodp matrix"
] | algebra/mxalgebra.v | center_mxP | |
mxring_id_uniqm n (R : 'A_(m, n)) e1 e2 :
mxring_id R e1 -> mxring_id R e2 -> e1 = e2.
Proof.
by case=> [_ Re1 idRe1 _] [_ Re2 _ ide2R]; rewrite -(idRe1 _ Re2) ide2R.
Qed. | Lemma | algebra | [
"From HB Require Import structures",
"From mathcomp Require Import ssreflect ssrbool ssrfun eqtype ssrnat seq choice",
"From mathcomp Require Import fintype finfun bigop finset fingroup perm order",
"From mathcomp Require Import div prime binomial ssralg finalg zmodp matrix"
] | algebra/mxalgebra.v | mxring_id_uniq | |
cent_mx_idealm n (R : 'A_(m, n)) : left_mx_ideal 'C(R)%MS 'C(R)%MS.
Proof.
apply/mulsmx_subP=> A1 A2 C_A1 C_A2; apply/cent_mxP=> B R_B.
by rewrite mulmxA (cent_mxP C_A1) // -!mulmxA (cent_mxP C_A2).
Qed. | Lemma | algebra | [
"From HB Require Import structures",
"From mathcomp Require Import ssreflect ssrbool ssrfun eqtype ssrnat seq choice",
"From mathcomp Require Import fintype finfun bigop finset fingroup perm order",
"From mathcomp Require Import div prime binomial ssralg finalg zmodp matrix"
] | algebra/mxalgebra.v | cent_mx_ideal | |
cent_mx_ringm n (R : 'A_(m, n)) : n > 0 -> mxring 'C(R)%MS.
Proof.
move=> n_gt0; rewrite /mxring cent_mx_ideal; apply/mxring_idP.
exists 1%:M; split=> [||A _|A _]; rewrite ?mulmx1 ?mul1mx ?scalar_mx_cent //.
by rewrite -mxrank_eq0 mxrank1 -lt0n.
Qed. | Lemma | algebra | [
"From HB Require Import structures",
"From mathcomp Require Import ssreflect ssrbool ssrfun eqtype ssrnat seq choice",
"From mathcomp Require Import fintype finfun bigop finset fingroup perm order",
"From mathcomp Require Import div prime binomial ssralg finalg zmodp matrix"
] | algebra/mxalgebra.v | cent_mx_ring | |
mxdirect_adds_centerm1 m2 n (R1 : 'A_(m1, n)) (R2 : 'A_(m2, n)) :
mx_ideal (R1 + R2)%MS R1 -> mx_ideal (R1 + R2)%MS R2 ->
mxdirect (R1 + R2) ->
('Z((R1 + R2)%MS) :=: 'Z(R1) + 'Z(R2))%MS.
Proof.
case/andP=> idlR1 idrR1 /andP[idlR2 idrR2] /mxdirect_addsP dxR12.
apply/eqmxP/andP; split.
apply/memmx_subP=> z0; ... | Lemma | algebra | [
"From HB Require Import structures",
"From mathcomp Require Import ssreflect ssrbool ssrfun eqtype ssrnat seq choice",
"From mathcomp Require Import fintype finfun bigop finset fingroup perm order",
"From mathcomp Require Import div prime binomial ssralg finalg zmodp matrix"
] | algebra/mxalgebra.v | mxdirect_adds_center | |
mxdirect_sums_center(I : finType) m n (R : 'A_(m, n)) R_ :
(\sum_i R_ i :=: R)%MS -> mxdirect (\sum_i R_ i) ->
(forall i : I, mx_ideal R (R_ i)) ->
('Z(R) :=: \sum_i 'Z(R_ i))%MS.
Proof.
move=> defR dxR idealR.
have sR_R: (R_ _ <= R)%MS by move=> i; rewrite -defR (sumsmx_sup i).
have anhR i j A B : i != j -> ... | Lemma | algebra | [
"From HB Require Import structures",
"From mathcomp Require Import ssreflect ssrbool ssrfun eqtype ssrnat seq choice",
"From mathcomp Require Import fintype finfun bigop finset fingroup perm order",
"From mathcomp Require Import div prime binomial ssralg finalg zmodp matrix"
] | algebra/mxalgebra.v | mxdirect_sums_center | |
Gaussian_elimination_mapm n (A : 'M_(m, n)) :
Gaussian_elimination_ A^f = ((col_ebase A)^f, (row_ebase A)^f, \rank A).
Proof.
rewrite mxrankE /row_ebase /col_ebase unlock.
elim: m n A => [|m IHm] [|n] A /=; rewrite ?map_mx1 //.
set pAnz := [pred k | A k.1 k.2 != 0].
rewrite (@eq_pick _ _ pAnz) => [|k]; last by rewrit... | Lemma | algebra | [
"From HB Require Import structures",
"From mathcomp Require Import ssreflect ssrbool ssrfun eqtype ssrnat seq choice",
"From mathcomp Require Import fintype finfun bigop finset fingroup perm order",
"From mathcomp Require Import div prime binomial ssralg finalg zmodp matrix"
] | algebra/mxalgebra.v | Gaussian_elimination_map | |
mxrank_mapm n (A : 'M_(m, n)) : \rank A^f = \rank A.
Proof. by rewrite mxrankE Gaussian_elimination_map. Qed. | Lemma | algebra | [
"From HB Require Import structures",
"From mathcomp Require Import ssreflect ssrbool ssrfun eqtype ssrnat seq choice",
"From mathcomp Require Import fintype finfun bigop finset fingroup perm order",
"From mathcomp Require Import div prime binomial ssralg finalg zmodp matrix"
] | algebra/mxalgebra.v | mxrank_map | |
row_free_mapm n (A : 'M_(m, n)) : row_free A^f = row_free A.
Proof. by rewrite /row_free mxrank_map. Qed. | Lemma | algebra | [
"From HB Require Import structures",
"From mathcomp Require Import ssreflect ssrbool ssrfun eqtype ssrnat seq choice",
"From mathcomp Require Import fintype finfun bigop finset fingroup perm order",
"From mathcomp Require Import div prime binomial ssralg finalg zmodp matrix"
] | algebra/mxalgebra.v | row_free_map | |
row_full_mapm n (A : 'M_(m, n)) : row_full A^f = row_full A.
Proof. by rewrite /row_full mxrank_map. Qed. | Lemma | algebra | [
"From HB Require Import structures",
"From mathcomp Require Import ssreflect ssrbool ssrfun eqtype ssrnat seq choice",
"From mathcomp Require Import fintype finfun bigop finset fingroup perm order",
"From mathcomp Require Import div prime binomial ssralg finalg zmodp matrix"
] | algebra/mxalgebra.v | row_full_map | |
map_row_ebasem n (A : 'M_(m, n)) : (row_ebase A)^f = row_ebase A^f.
Proof. by rewrite {2}/row_ebase unlock Gaussian_elimination_map. Qed. | Lemma | algebra | [
"From HB Require Import structures",
"From mathcomp Require Import ssreflect ssrbool ssrfun eqtype ssrnat seq choice",
"From mathcomp Require Import fintype finfun bigop finset fingroup perm order",
"From mathcomp Require Import div prime binomial ssralg finalg zmodp matrix"
] | algebra/mxalgebra.v | map_row_ebase | |
map_col_ebasem n (A : 'M_(m, n)) : (col_ebase A)^f = col_ebase A^f.
Proof. by rewrite {2}/col_ebase unlock Gaussian_elimination_map. Qed. | Lemma | algebra | [
"From HB Require Import structures",
"From mathcomp Require Import ssreflect ssrbool ssrfun eqtype ssrnat seq choice",
"From mathcomp Require Import fintype finfun bigop finset fingroup perm order",
"From mathcomp Require Import div prime binomial ssralg finalg zmodp matrix"
] | algebra/mxalgebra.v | map_col_ebase | |
map_row_basem n (A : 'M_(m, n)) :
(row_base A)^f = castmx (mxrank_map A, erefl n) (row_base A^f).
Proof.
move: (mxrank_map A); rewrite {2}/row_base mxrank_map => eqrr.
by rewrite castmx_id map_mxM map_pid_mx map_row_ebase.
Qed. | Lemma | algebra | [
"From HB Require Import structures",
"From mathcomp Require Import ssreflect ssrbool ssrfun eqtype ssrnat seq choice",
"From mathcomp Require Import fintype finfun bigop finset fingroup perm order",
"From mathcomp Require Import div prime binomial ssralg finalg zmodp matrix"
] | algebra/mxalgebra.v | map_row_base | |
map_col_basem n (A : 'M_(m, n)) :
(col_base A)^f = castmx (erefl m, mxrank_map A) (col_base A^f).
Proof.
move: (mxrank_map A); rewrite {2}/col_base mxrank_map => eqrr.
by rewrite castmx_id map_mxM map_pid_mx map_col_ebase.
Qed. | Lemma | algebra | [
"From HB Require Import structures",
"From mathcomp Require Import ssreflect ssrbool ssrfun eqtype ssrnat seq choice",
"From mathcomp Require Import fintype finfun bigop finset fingroup perm order",
"From mathcomp Require Import div prime binomial ssralg finalg zmodp matrix"
] | algebra/mxalgebra.v | map_col_base | |
map_pinvmxm n (A : 'M_(m, n)) : (pinvmx A)^f = pinvmx A^f.
Proof.
rewrite !map_mxM !map_invmx map_row_ebase map_col_ebase.
by rewrite map_pid_mx -mxrank_map.
Qed. | Lemma | algebra | [
"From HB Require Import structures",
"From mathcomp Require Import ssreflect ssrbool ssrfun eqtype ssrnat seq choice",
"From mathcomp Require Import fintype finfun bigop finset fingroup perm order",
"From mathcomp Require Import div prime binomial ssralg finalg zmodp matrix"
] | algebra/mxalgebra.v | map_pinvmx | |
map_kermxm n (A : 'M_(m, n)) : (kermx A)^f = kermx A^f.
Proof.
by rewrite !map_mxM map_invmx map_col_ebase -mxrank_map map_copid_mx.
Qed. | Lemma | algebra | [
"From HB Require Import structures",
"From mathcomp Require Import ssreflect ssrbool ssrfun eqtype ssrnat seq choice",
"From mathcomp Require Import fintype finfun bigop finset fingroup perm order",
"From mathcomp Require Import div prime binomial ssralg finalg zmodp matrix"
] | algebra/mxalgebra.v | map_kermx | |
map_cokermxm n (A : 'M_(m, n)) : (cokermx A)^f = cokermx A^f.
Proof.
by rewrite !map_mxM map_invmx map_row_ebase -mxrank_map map_copid_mx.
Qed. | Lemma | algebra | [
"From HB Require Import structures",
"From mathcomp Require Import ssreflect ssrbool ssrfun eqtype ssrnat seq choice",
"From mathcomp Require Import fintype finfun bigop finset fingroup perm order",
"From mathcomp Require Import div prime binomial ssralg finalg zmodp matrix"
] | algebra/mxalgebra.v | map_cokermx | |
map_submxm1 m2 n (A : 'M_(m1, n)) (B : 'M_(m2, n)) :
(A^f <= B^f)%MS = (A <= B)%MS.
Proof. by rewrite !submxE -map_cokermx -map_mxM map_mx_eq0. Qed. | Lemma | algebra | [
"From HB Require Import structures",
"From mathcomp Require Import ssreflect ssrbool ssrfun eqtype ssrnat seq choice",
"From mathcomp Require Import fintype finfun bigop finset fingroup perm order",
"From mathcomp Require Import div prime binomial ssralg finalg zmodp matrix"
] | algebra/mxalgebra.v | map_submx | |
map_ltmxm1 m2 n (A : 'M_(m1, n)) (B : 'M_(m2, n)) :
(A^f < B^f)%MS = (A < B)%MS.
Proof. by rewrite /ltmx !map_submx. Qed. | Lemma | algebra | [
"From HB Require Import structures",
"From mathcomp Require Import ssreflect ssrbool ssrfun eqtype ssrnat seq choice",
"From mathcomp Require Import fintype finfun bigop finset fingroup perm order",
"From mathcomp Require Import div prime binomial ssralg finalg zmodp matrix"
] | algebra/mxalgebra.v | map_ltmx | |
map_eqmxm1 m2 n (A : 'M_(m1, n)) (B : 'M_(m2, n)) :
(A^f :=: B^f)%MS <-> (A :=: B)%MS.
Proof.
split=> [/eqmxP|eqAB]; first by rewrite !map_submx => /eqmxP.
by apply/eqmxP; rewrite !map_submx !eqAB !submx_refl.
Qed. | Lemma | algebra | [
"From HB Require Import structures",
"From mathcomp Require Import ssreflect ssrbool ssrfun eqtype ssrnat seq choice",
"From mathcomp Require Import fintype finfun bigop finset fingroup perm order",
"From mathcomp Require Import div prime binomial ssralg finalg zmodp matrix"
] | algebra/mxalgebra.v | map_eqmx | |
map_genmxm n (A : 'M_(m, n)) : (<<A>>^f :=: <<A^f>>)%MS.
Proof. by apply/eqmxP; rewrite !(genmxE, map_submx) andbb. Qed. | Lemma | algebra | [
"From HB Require Import structures",
"From mathcomp Require Import ssreflect ssrbool ssrfun eqtype ssrnat seq choice",
"From mathcomp Require Import fintype finfun bigop finset fingroup perm order",
"From mathcomp Require Import div prime binomial ssralg finalg zmodp matrix"
] | algebra/mxalgebra.v | map_genmx | |
map_addsmxm1 m2 n (A : 'M_(m1, n)) (B : 'M_(m2, n)) :
(((A + B)%MS)^f :=: A^f + B^f)%MS.
Proof.
by apply/eqmxP; rewrite !addsmxE -map_col_mx !map_submx !addsmxE andbb.
Qed. | Lemma | algebra | [
"From HB Require Import structures",
"From mathcomp Require Import ssreflect ssrbool ssrfun eqtype ssrnat seq choice",
"From mathcomp Require Import fintype finfun bigop finset fingroup perm order",
"From mathcomp Require Import div prime binomial ssralg finalg zmodp matrix"
] | algebra/mxalgebra.v | map_addsmx | |
map_capmx_genm1 m2 n (A : 'M_(m1, n)) (B : 'M_(m2, n)) :
(capmx_gen A B)^f = capmx_gen A^f B^f.
Proof. by rewrite map_mxM map_lsubmx map_kermx map_col_mx. Qed. | Lemma | algebra | [
"From HB Require Import structures",
"From mathcomp Require Import ssreflect ssrbool ssrfun eqtype ssrnat seq choice",
"From mathcomp Require Import fintype finfun bigop finset fingroup perm order",
"From mathcomp Require Import div prime binomial ssralg finalg zmodp matrix"
] | algebra/mxalgebra.v | map_capmx_gen | |
map_capmxm1 m2 n (A : 'M_(m1, n)) (B : 'M_(m2, n)) :
((A :&: B)^f :=: A^f :&: B^f)%MS.
Proof.
by apply/eqmxP; rewrite !capmxE -map_capmx_gen !map_submx -!capmxE andbb.
Qed. | Lemma | algebra | [
"From HB Require Import structures",
"From mathcomp Require Import ssreflect ssrbool ssrfun eqtype ssrnat seq choice",
"From mathcomp Require Import fintype finfun bigop finset fingroup perm order",
"From mathcomp Require Import div prime binomial ssralg finalg zmodp matrix"
] | algebra/mxalgebra.v | map_capmx | |
map_complmxm n (A : 'M_(m, n)) : (A^C^f = A^f^C)%MS.
Proof. by rewrite map_mxM map_row_ebase -mxrank_map map_copid_mx. Qed. | Lemma | algebra | [
"From HB Require Import structures",
"From mathcomp Require Import ssreflect ssrbool ssrfun eqtype ssrnat seq choice",
"From mathcomp Require Import fintype finfun bigop finset fingroup perm order",
"From mathcomp Require Import div prime binomial ssralg finalg zmodp matrix"
] | algebra/mxalgebra.v | map_complmx | |
map_diffmxm1 m2 n (A : 'M_(m1, n)) (B : 'M_(m2, n)) :
((A :\: B)^f :=: A^f :\: B^f)%MS.
Proof.
apply/eqmxP; rewrite !diffmxE -map_capmx_gen -map_complmx.
by rewrite -!map_capmx !map_submx -!diffmxE andbb.
Qed. | Lemma | algebra | [
"From HB Require Import structures",
"From mathcomp Require Import ssreflect ssrbool ssrfun eqtype ssrnat seq choice",
"From mathcomp Require Import fintype finfun bigop finset fingroup perm order",
"From mathcomp Require Import div prime binomial ssralg finalg zmodp matrix"
] | algebra/mxalgebra.v | map_diffmx | |
map_eigenspacen (g : 'M_n) a : (eigenspace g a)^f = eigenspace g^f (f a).
Proof. by rewrite map_kermx map_mxB ?map_scalar_mx. Qed. | Lemma | algebra | [
"From HB Require Import structures",
"From mathcomp Require Import ssreflect ssrbool ssrfun eqtype ssrnat seq choice",
"From mathcomp Require Import fintype finfun bigop finset fingroup perm order",
"From mathcomp Require Import div prime binomial ssralg finalg zmodp matrix"
] | algebra/mxalgebra.v | map_eigenspace | |
eigenvalue_mapn (g : 'M_n) a : eigenvalue g^f (f a) = eigenvalue g a.
Proof. by rewrite /eigenvalue -map_eigenspace map_mx_eq0. Qed. | Lemma | algebra | [
"From HB Require Import structures",
"From mathcomp Require Import ssreflect ssrbool ssrfun eqtype ssrnat seq choice",
"From mathcomp Require Import fintype finfun bigop finset fingroup perm order",
"From mathcomp Require Import div prime binomial ssralg finalg zmodp matrix"
] | algebra/mxalgebra.v | eigenvalue_map | |
memmx_mapm n A (E : 'A_(m, n)) : (A^f \in E^f)%MS = (A \in E)%MS.
Proof. by rewrite -map_mxvec map_submx. Qed. | Lemma | algebra | [
"From HB Require Import structures",
"From mathcomp Require Import ssreflect ssrbool ssrfun eqtype ssrnat seq choice",
"From mathcomp Require Import fintype finfun bigop finset fingroup perm order",
"From mathcomp Require Import div prime binomial ssralg finalg zmodp matrix"
] | algebra/mxalgebra.v | memmx_map | |
map_mulsmxm1 m2 n (E1 : 'A_(m1, n)) (E2 : 'A_(m2, n)) :
((E1 * E2)%MS^f :=: E1^f * E2^f)%MS.
Proof.
rewrite /mulsmx; elim/big_rec2: _ => [|i A Af _ eqA]; first by rewrite map_mx0.
apply: (eqmx_trans (map_addsmx _ _)); apply: adds_eqmx {A Af}eqA.
apply/eqmxP; rewrite !map_genmx !genmxE map_mxM.
apply/rV_eqP=> u; congr... | Lemma | algebra | [
"From HB Require Import structures",
"From mathcomp Require Import ssreflect ssrbool ssrfun eqtype ssrnat seq choice",
"From mathcomp Require Import fintype finfun bigop finset fingroup perm order",
"From mathcomp Require Import div prime binomial ssralg finalg zmodp matrix"
] | algebra/mxalgebra.v | map_mulsmx | |
map_cent_mxm n (E : 'A_(m, n)) : ('C(E)%MS)^f = 'C(E^f)%MS.
Proof.
rewrite map_kermx; congr kermx; apply: map_lin_mx => A; rewrite map_mxM.
by congr (_ *m _); apply: map_lin_mx => B; rewrite map_mxB ?map_mxM.
Qed. | Lemma | algebra | [
"From HB Require Import structures",
"From mathcomp Require Import ssreflect ssrbool ssrfun eqtype ssrnat seq choice",
"From mathcomp Require Import fintype finfun bigop finset fingroup perm order",
"From mathcomp Require Import div prime binomial ssralg finalg zmodp matrix"
] | algebra/mxalgebra.v | map_cent_mx | |
map_center_mxm n (E : 'A_(m, n)) : (('Z(E))^f :=: 'Z(E^f))%MS.
Proof. by rewrite /center_mx -map_cent_mx; apply: map_capmx. Qed. | Lemma | algebra | [
"From HB Require Import structures",
"From mathcomp Require Import ssreflect ssrbool ssrfun eqtype ssrnat seq choice",
"From mathcomp Require Import fintype finfun bigop finset fingroup perm order",
"From mathcomp Require Import div prime binomial ssralg finalg zmodp matrix"
] | algebra/mxalgebra.v | map_center_mx | |
eqmx_col{m} (V_ : forall i, 'M[F]_(p_ i, m)) :
(\mxcol_i V_ i :=: \sum_i <<V_ i>>)%MS.
Proof.
apply/eqmxP/andP; split.
apply/row_subP => i; rewrite row_mxcol.
by rewrite (sumsmx_sup (sig1 i))// genmxE row_sub.
apply/sumsmx_subP => i0 _; rewrite genmxE; apply/row_subP => j.
apply: (eq_row_sub (Rank _ j)); apply/ro... | Lemma | algebra | [
"From HB Require Import structures",
"From mathcomp Require Import ssreflect ssrbool ssrfun eqtype ssrnat seq choice",
"From mathcomp Require Import fintype finfun bigop finset fingroup perm order",
"From mathcomp Require Import div prime binomial ssralg finalg zmodp matrix"
] | algebra/mxalgebra.v | eqmx_col | |
rank_mxdiag(V_ : forall i, 'M[F]_(p_ i)) :
(\rank (\mxdiag_i V_ i) = \sum_i \rank (V_ i))%N.
Proof.
elim: {+}n {+}p_ V_ => [|m IHm] q_ V_.
by move: (\mxdiag__ _); rewrite !big_ord0 => M; rewrite flatmx0 mxrank0.
rewrite mxdiag_recl [RHS]big_ord_recl/= -IHm.
by case: _ / mxsize_recl; rewrite ?castmx_id rank_diag_blo... | Lemma | algebra | [
"From HB Require Import structures",
"From mathcomp Require Import ssreflect ssrbool ssrfun eqtype ssrnat seq choice",
"From mathcomp Require Import fintype finfun bigop finset fingroup perm order",
"From mathcomp Require Import div prime binomial ssralg finalg zmodp matrix"
] | algebra/mxalgebra.v | rank_mxdiag | |
rVpolyv := \poly_(k < d) (if insub k is Some i then v 0 i else 0). | Definition | algebra | [
"From HB Require Import structures",
"From mathcomp Require Import ssreflect ssrfun ssrbool eqtype ssrnat choice seq",
"From mathcomp Require Import div fintype tuple finfun bigop fingroup perm",
"From mathcomp Require Import ssralg zmodp matrix mxalgebra poly polydiv"
] | algebra/mxpoly.v | rVpoly | |
poly_rVp := \row_(i < d) p`_i. | Definition | algebra | [
"From HB Require Import structures",
"From mathcomp Require Import ssreflect ssrfun ssrbool eqtype ssrnat choice seq",
"From mathcomp Require Import div fintype tuple finfun bigop fingroup perm",
"From mathcomp Require Import ssralg zmodp matrix mxalgebra poly polydiv"
] | algebra/mxpoly.v | poly_rV | |
coef_rVpolyv k : (rVpoly v)`_k = if insub k is Some i then v 0 i else 0.
Proof. by rewrite coef_poly; case: insubP => [i ->|]; rewrite ?if_same. Qed. | Lemma | algebra | [
"From HB Require Import structures",
"From mathcomp Require Import ssreflect ssrfun ssrbool eqtype ssrnat choice seq",
"From mathcomp Require Import div fintype tuple finfun bigop fingroup perm",
"From mathcomp Require Import ssralg zmodp matrix mxalgebra poly polydiv"
] | algebra/mxpoly.v | coef_rVpoly | |
coef_rVpoly_ordv (i : 'I_d) : (rVpoly v)`_i = v 0 i.
Proof. by rewrite coef_rVpoly valK. Qed. | Lemma | algebra | [
"From HB Require Import structures",
"From mathcomp Require Import ssreflect ssrfun ssrbool eqtype ssrnat choice seq",
"From mathcomp Require Import div fintype tuple finfun bigop fingroup perm",
"From mathcomp Require Import ssralg zmodp matrix mxalgebra poly polydiv"
] | algebra/mxpoly.v | coef_rVpoly_ord | |
rVpoly_deltai : rVpoly (delta_mx 0 i) = 'X^i.
Proof.
apply/polyP=> j; rewrite coef_rVpoly coefXn.
case: insubP => [k _ <- | j_ge_d]; first by rewrite mxE.
by case: eqP j_ge_d => // ->; rewrite ltn_ord.
Qed. | Lemma | algebra | [
"From HB Require Import structures",
"From mathcomp Require Import ssreflect ssrfun ssrbool eqtype ssrnat choice seq",
"From mathcomp Require Import div fintype tuple finfun bigop fingroup perm",
"From mathcomp Require Import ssralg zmodp matrix mxalgebra poly polydiv"
] | algebra/mxpoly.v | rVpoly_delta | |
rVpolyK: cancel rVpoly poly_rV.
Proof. by move=> u; apply/rowP=> i; rewrite mxE coef_rVpoly_ord. Qed. | Lemma | algebra | [
"From HB Require Import structures",
"From mathcomp Require Import ssreflect ssrfun ssrbool eqtype ssrnat choice seq",
"From mathcomp Require Import div fintype tuple finfun bigop fingroup perm",
"From mathcomp Require Import ssralg zmodp matrix mxalgebra poly polydiv"
] | algebra/mxpoly.v | rVpolyK | |
poly_rV_Kp : size p <= d -> rVpoly (poly_rV p) = p.
Proof.
move=> le_p_d; apply/polyP=> k; rewrite coef_rVpoly.
case: insubP => [i _ <- | ]; first by rewrite mxE.
by rewrite -ltnNge => le_d_l; rewrite nth_default ?(leq_trans le_p_d).
Qed. | Lemma | algebra | [
"From HB Require Import structures",
"From mathcomp Require Import ssreflect ssrfun ssrbool eqtype ssrnat choice seq",
"From mathcomp Require Import div fintype tuple finfun bigop fingroup perm",
"From mathcomp Require Import ssralg zmodp matrix mxalgebra poly polydiv"
] | algebra/mxpoly.v | poly_rV_K | |
poly_rV_is_semilinear: semilinear poly_rV.
Proof. by split=> [a p|p q]; apply/rowP=> i; rewrite !mxE (coefZ, coefD). Qed.
HB.instance Definition _ := GRing.isSemilinear.Build R {poly R} 'rV_d _ poly_rV
poly_rV_is_semilinear.
#[deprecated(since="mathcomp 2.5.0", note="Use linearP instead.")] | Lemma | algebra | [
"From HB Require Import structures",
"From mathcomp Require Import ssreflect ssrfun ssrbool eqtype ssrnat choice seq",
"From mathcomp Require Import div fintype tuple finfun bigop fingroup perm",
"From mathcomp Require Import ssralg zmodp matrix mxalgebra poly polydiv"
] | algebra/mxpoly.v | poly_rV_is_semilinear | |
poly_rV_is_linear: linear poly_rV. Proof. exact: linearP. Qed. | Lemma | algebra | [
"From HB Require Import structures",
"From mathcomp Require Import ssreflect ssrfun ssrbool eqtype ssrnat choice seq",
"From mathcomp Require Import div fintype tuple finfun bigop fingroup perm",
"From mathcomp Require Import ssralg zmodp matrix mxalgebra poly polydiv"
] | algebra/mxpoly.v | poly_rV_is_linear | |
rVpoly_is_semilinear: semilinear rVpoly.
Proof.
split=> [a u|u v]; apply/polyP=> k; rewrite (coefZ, coefD) !coef_rVpoly.
by case: insubP => [i _ _|_]; rewrite ?mxE // mulr0.
by case: insubP=> [i _ _|_]; rewrite ?mxE ?addr0.
Qed.
HB.instance Definition _ := GRing.isSemilinear.Build R 'rV_d {poly R} _ rVpoly
rVpoly_i... | Lemma | algebra | [
"From HB Require Import structures",
"From mathcomp Require Import ssreflect ssrfun ssrbool eqtype ssrnat choice seq",
"From mathcomp Require Import div fintype tuple finfun bigop fingroup perm",
"From mathcomp Require Import ssralg zmodp matrix mxalgebra poly polydiv"
] | algebra/mxpoly.v | rVpoly_is_semilinear | |
rvPoly_is_linear: linear rVpoly. Proof. exact: linearP. Qed. | Lemma | algebra | [
"From HB Require Import structures",
"From mathcomp Require Import ssreflect ssrfun ssrbool eqtype ssrnat choice seq",
"From mathcomp Require Import div fintype tuple finfun bigop fingroup perm",
"From mathcomp Require Import ssralg zmodp matrix mxalgebra poly polydiv"
] | algebra/mxpoly.v | rvPoly_is_linear | |
Sylvester_mx: 'M[R]_dS := col_mx (band p) (band q). | Definition | algebra | [
"From HB Require Import structures",
"From mathcomp Require Import ssreflect ssrfun ssrbool eqtype ssrnat choice seq",
"From mathcomp Require Import div fintype tuple finfun bigop fingroup perm",
"From mathcomp Require Import ssralg zmodp matrix mxalgebra poly polydiv"
] | algebra/mxpoly.v | Sylvester_mx | |
Sylvester_mxE(i j : 'I_dS) :
let S_ r k := r`_(j - k) *+ (k <= j) in
Sylvester_mx i j = match split i with inl k => S_ p k | inr k => S_ q k end.
Proof.
move=> S_ /[1!mxE]; case: {i}(split i) => i /[!mxE]/=;
by rewrite rVpoly_delta coefXnM ltnNge if_neg -mulrb.
Qed. | Lemma | algebra | [
"From HB Require Import structures",
"From mathcomp Require Import ssreflect ssrfun ssrbool eqtype ssrnat choice seq",
"From mathcomp Require Import div fintype tuple finfun bigop fingroup perm",
"From mathcomp Require Import ssralg zmodp matrix mxalgebra poly polydiv"
] | algebra/mxpoly.v | Sylvester_mxE | |
resultant:= \det Sylvester_mx. | Definition | algebra | [
"From HB Require Import structures",
"From mathcomp Require Import ssreflect ssrfun ssrbool eqtype ssrnat choice seq",
"From mathcomp Require Import div fintype tuple finfun bigop fingroup perm",
"From mathcomp Require Import ssralg zmodp matrix mxalgebra poly polydiv"
] | algebra/mxpoly.v | resultant | |
resultant_in_ideal(R : comNzRingType) (p q : {poly R}) :
size p > 1 -> size q > 1 ->
{uv : {poly R} * {poly R} | size uv.1 < size q /\ size uv.2 < size p
& (resultant p q)%:P = uv.1 * p + uv.2 * q}.
Proof.
move=> p_nc q_nc; pose dp := (size p).-1; pose dq := (size q).-1.
pose S := Sylvester_mx p q; pose dS := (... | Lemma | algebra | [
"From HB Require Import structures",
"From mathcomp Require Import ssreflect ssrfun ssrbool eqtype ssrnat choice seq",
"From mathcomp Require Import div fintype tuple finfun bigop fingroup perm",
"From mathcomp Require Import ssralg zmodp matrix mxalgebra poly polydiv"
] | algebra/mxpoly.v | resultant_in_ideal | |
resultant_eq0(R : idomainType) (p q : {poly R}) :
(resultant p q == 0) = (size (gcdp p q) > 1).
Proof.
have dvdpp := dvdpp; set r := gcdp p q.
pose dp := (size p).-1; pose dq := (size q).-1.
have /andP[r_p r_q]: (r %| p) && (r %| q) by rewrite -dvdp_gcd.
apply/det0P/idP=> [[uv nz_uv] | r_nonC].
have [p0 _ | p_nz] :... | Lemma | algebra | [
"From HB Require Import structures",
"From mathcomp Require Import ssreflect ssrfun ssrbool eqtype ssrnat choice seq",
"From mathcomp Require Import div fintype tuple finfun bigop fingroup perm",
"From mathcomp Require Import ssralg zmodp matrix mxalgebra poly polydiv"
] | algebra/mxpoly.v | resultant_eq0 | |
horner_mx:= horner_morph (comm_mx_scalar^~ A).
HB.instance Definition _ := GRing.RMorphism.on horner_mx. | Definition | algebra | [
"From HB Require Import structures",
"From mathcomp Require Import ssreflect ssrfun ssrbool eqtype ssrnat choice seq",
"From mathcomp Require Import div fintype tuple finfun bigop fingroup perm",
"From mathcomp Require Import ssralg zmodp matrix mxalgebra poly polydiv"
] | algebra/mxpoly.v | horner_mx | |
horner_mx_Ca : horner_mx a%:P = a%:M.
Proof. exact: horner_morphC. Qed. | Lemma | algebra | [
"From HB Require Import structures",
"From mathcomp Require Import ssreflect ssrfun ssrbool eqtype ssrnat choice seq",
"From mathcomp Require Import div fintype tuple finfun bigop fingroup perm",
"From mathcomp Require Import ssralg zmodp matrix mxalgebra poly polydiv"
] | algebra/mxpoly.v | horner_mx_C | |
horner_mx_X: horner_mx 'X = A. Proof. exact: horner_morphX. Qed. | Lemma | algebra | [
"From HB Require Import structures",
"From mathcomp Require Import ssreflect ssrfun ssrbool eqtype ssrnat choice seq",
"From mathcomp Require Import div fintype tuple finfun bigop fingroup perm",
"From mathcomp Require Import ssralg zmodp matrix mxalgebra poly polydiv"
] | algebra/mxpoly.v | horner_mx_X | |
horner_mxZ: scalable horner_mx.
Proof.
move=> a p /=; rewrite -mul_polyC rmorphM /=.
by rewrite horner_mx_C [_ * _]mul_scalar_mx.
Qed.
HB.instance Definition _ := GRing.isScalable.Build R _ _ *:%R horner_mx
horner_mxZ. | Lemma | algebra | [
"From HB Require Import structures",
"From mathcomp Require Import ssreflect ssrfun ssrbool eqtype ssrnat choice seq",
"From mathcomp Require Import div fintype tuple finfun bigop fingroup perm",
"From mathcomp Require Import ssralg zmodp matrix mxalgebra poly polydiv"
] | algebra/mxpoly.v | horner_mxZ | |
powers_mxd := \matrix_(i < d) mxvec (A ^+ i). | Definition | algebra | [
"From HB Require Import structures",
"From mathcomp Require Import ssreflect ssrfun ssrbool eqtype ssrnat choice seq",
"From mathcomp Require Import div fintype tuple finfun bigop fingroup perm",
"From mathcomp Require Import ssralg zmodp matrix mxalgebra poly polydiv"
] | algebra/mxpoly.v | powers_mx | |
horner_rVpolym (u : 'rV_m) :
horner_mx (rVpoly u) = vec_mx (u *m powers_mx m).
Proof.
rewrite mulmx_sum_row [rVpoly u]poly_def 2!linear_sum; apply: eq_bigr => i _.
by rewrite valK /= 2!linearZ rmorphXn/= horner_mx_X rowK mxvecK.
Qed. | Lemma | algebra | [
"From HB Require Import structures",
"From mathcomp Require Import ssreflect ssrfun ssrbool eqtype ssrnat choice seq",
"From mathcomp Require Import div fintype tuple finfun bigop fingroup perm",
"From mathcomp Require Import ssralg zmodp matrix mxalgebra poly polydiv"
] | algebra/mxpoly.v | horner_rVpoly | |
horner_mx_diag(d : 'rV[R]_n) (p : {poly R}) :
horner_mx (diag_mx d) p = diag_mx (map_mx (horner p) d).
Proof.
apply/matrixP => i j; rewrite !mxE.
elim/poly_ind: p => [|p c ihp]; first by rewrite rmorph0 horner0 mxE mul0rn.
rewrite !hornerE mulrnDl rmorphD rmorphM /= horner_mx_X horner_mx_C !mxE.
rewrite (bigD1 j)//= ... | Lemma | algebra | [
"From HB Require Import structures",
"From mathcomp Require Import ssreflect ssrfun ssrbool eqtype ssrnat choice seq",
"From mathcomp Require Import div fintype tuple finfun bigop fingroup perm",
"From mathcomp Require Import ssralg zmodp matrix mxalgebra poly polydiv"
] | algebra/mxpoly.v | horner_mx_diag | |
comm_mx_hornerA B p : comm_mx A B -> comm_mx A (horner_mx B p).
Proof.
move=> fg; apply: commr_horner => // i.
by rewrite coef_map; apply/comm_scalar_mx.
Qed. | Lemma | algebra | [
"From HB Require Import structures",
"From mathcomp Require Import ssreflect ssrfun ssrbool eqtype ssrnat choice seq",
"From mathcomp Require Import div fintype tuple finfun bigop fingroup perm",
"From mathcomp Require Import ssralg zmodp matrix mxalgebra poly polydiv"
] | algebra/mxpoly.v | comm_mx_horner | |
comm_horner_mxA B p : comm_mx A B -> comm_mx (horner_mx A p) B.
Proof. by move=> ?; apply/comm_mx_sym/comm_mx_horner/comm_mx_sym. Qed. | Lemma | algebra | [
"From HB Require Import structures",
"From mathcomp Require Import ssreflect ssrfun ssrbool eqtype ssrnat choice seq",
"From mathcomp Require Import div fintype tuple finfun bigop fingroup perm",
"From mathcomp Require Import ssralg zmodp matrix mxalgebra poly polydiv"
] | algebra/mxpoly.v | comm_horner_mx | |
comm_horner_mx2A p q : GRing.comm (horner_mx A p) (horner_mx A q).
Proof. exact/comm_mx_horner/comm_horner_mx. Qed. | Lemma | algebra | [
"From HB Require Import structures",
"From mathcomp Require Import ssreflect ssrfun ssrbool eqtype ssrnat choice seq",
"From mathcomp Require Import div fintype tuple finfun bigop fingroup perm",
"From mathcomp Require Import ssralg zmodp matrix mxalgebra poly polydiv"
] | algebra/mxpoly.v | comm_horner_mx2 | |
horner_mx_stable(K : fieldType) m n p
(V : 'M[K]_(n.+1, m.+1)) (f : 'M_m.+1) :
stablemx V f -> stablemx V (horner_mx f p).
Proof.
move=> V_fstab; elim/poly_ind: p => [|p c]; first by rewrite rmorph0 stablemx0.
move=> fp_stable; rewrite rmorphD rmorphM/= horner_mx_X horner_mx_C.
by rewrite stablemxD ?stablemxM ?fp... | Lemma | algebra | [
"From HB Require Import structures",
"From mathcomp Require Import ssreflect ssrfun ssrbool eqtype ssrnat choice seq",
"From mathcomp Require Import div fintype tuple finfun bigop fingroup perm",
"From mathcomp Require Import ssralg zmodp matrix mxalgebra poly polydiv"
] | algebra/mxpoly.v | horner_mx_stable | |
char_poly_mx:= 'X%:M - map_mx (@polyC R) A. | Definition | algebra | [
"From HB Require Import structures",
"From mathcomp Require Import ssreflect ssrfun ssrbool eqtype ssrnat choice seq",
"From mathcomp Require Import div fintype tuple finfun bigop fingroup perm",
"From mathcomp Require Import ssralg zmodp matrix mxalgebra poly polydiv"
] | algebra/mxpoly.v | char_poly_mx | |
char_poly:= \det char_poly_mx.
Let diagA := [seq A i i | i <- index_enum _ & true].
Let size_diagA : size diagA = n.
Proof. by rewrite -[n]card_ord size_map; have [e _ _ []] := big_enumP. Qed.
Let split_diagA :
exists2 q, \prod_(x <- diagA) ('X - x%:P) + q = char_poly & size q <= n.-1.
Proof.
rewrite [char_poly](bigD... | Definition | algebra | [
"From HB Require Import structures",
"From mathcomp Require Import ssreflect ssrfun ssrbool eqtype ssrnat choice seq",
"From mathcomp Require Import div fintype tuple finfun bigop fingroup perm",
"From mathcomp Require Import ssralg zmodp matrix mxalgebra poly polydiv"
] | algebra/mxpoly.v | char_poly | |
size_char_poly: size char_poly = n.+1.
Proof.
have [q <- lt_q_n] := split_diagA; have le_q_n := leq_trans lt_q_n (leq_pred n).
by rewrite size_polyDl size_prod_XsubC size_diagA.
Qed. | Lemma | algebra | [
"From HB Require Import structures",
"From mathcomp Require Import ssreflect ssrfun ssrbool eqtype ssrnat choice seq",
"From mathcomp Require Import div fintype tuple finfun bigop fingroup perm",
"From mathcomp Require Import ssralg zmodp matrix mxalgebra poly polydiv"
] | algebra/mxpoly.v | size_char_poly | |
char_poly_monic: char_poly \is monic.
Proof.
rewrite monicE -(monicP (monic_prod_XsubC diagA xpredT id)).
rewrite !lead_coefE size_char_poly.
have [q <- lt_q_n] := split_diagA; have le_q_n := leq_trans lt_q_n (leq_pred n).
by rewrite size_prod_XsubC size_diagA coefD (nth_default 0 le_q_n) addr0.
Qed. | Lemma | algebra | [
"From HB Require Import structures",
"From mathcomp Require Import ssreflect ssrfun ssrbool eqtype ssrnat choice seq",
"From mathcomp Require Import div fintype tuple finfun bigop fingroup perm",
"From mathcomp Require Import ssralg zmodp matrix mxalgebra poly polydiv"
] | algebra/mxpoly.v | char_poly_monic | |
char_poly_trace: n > 0 -> char_poly`_n.-1 = - \tr A.
Proof.
move=> n_gt0; have [q <- lt_q_n] := split_diagA; set p := \prod_(x <- _) _.
rewrite coefD {q lt_q_n}(nth_default 0 lt_q_n) addr0.
have{n_gt0} ->: p`_n.-1 = ('X * p)`_n by rewrite coefXM eqn0Ngt n_gt0.
have ->: \tr A = \sum_(x <- diagA) x by rewrite big_map big... | Lemma | algebra | [
"From HB Require Import structures",
"From mathcomp Require Import ssreflect ssrfun ssrbool eqtype ssrnat choice seq",
"From mathcomp Require Import div fintype tuple finfun bigop fingroup perm",
"From mathcomp Require Import ssralg zmodp matrix mxalgebra poly polydiv"
] | algebra/mxpoly.v | char_poly_trace | |
char_poly_det: char_poly`_0 = (- 1) ^+ n * \det A.
Proof.
rewrite big_distrr coef_sum [0%N]lock /=; apply: eq_bigr => s _.
rewrite -{1}rmorphN -rmorphXn mul_polyC coefZ /=.
rewrite mulrA -exprD addnC exprD -mulrA -lock; congr (_ * _).
transitivity (\prod_(i < n) - A i (s i)); last by rewrite prodrN card_ord.
elim: (ind... | Lemma | algebra | [
"From HB Require Import structures",
"From mathcomp Require Import ssreflect ssrfun ssrbool eqtype ssrnat choice seq",
"From mathcomp Require Import div fintype tuple finfun bigop fingroup perm",
"From mathcomp Require Import ssralg zmodp matrix mxalgebra poly polydiv"
] | algebra/mxpoly.v | char_poly_det | |
mx_poly_ring_isom(R : nzSemiRingType) n' (n := n'.+1) :
exists phi : {rmorphism 'M[{poly R}]_n -> {poly 'M[R]_n}},
[/\ bijective phi,
forall p, phi p%:M = map_poly scalar_mx p,
forall A, phi (map_mx polyC A) = A%:P
& forall A i j k, (phi A)`_k i j = (A i j)`_k].
Proof.
set M_RX := 'M[{poly R}]_n; se... | Lemma | algebra | [
"From HB Require Import structures",
"From mathcomp Require Import ssreflect ssrfun ssrbool eqtype ssrnat choice seq",
"From mathcomp Require Import div fintype tuple finfun bigop fingroup perm",
"From mathcomp Require Import ssralg zmodp matrix mxalgebra poly polydiv"
] | algebra/mxpoly.v | mx_poly_ring_isom | |
Cayley_Hamilton(R : comNzRingType) n' (A : 'M[R]_n'.+1) :
horner_mx A (char_poly A) = 0.
Proof.
have [phi [_ phiZ phiC _]] := mx_poly_ring_isom R n'.
apply/rootP/factor_theorem; rewrite -phiZ -mul_adj_mx rmorphM /=.
by move: (phi _) => q; exists q; rewrite rmorphB phiC phiZ map_polyX.
Qed. | Theorem | algebra | [
"From HB Require Import structures",
"From mathcomp Require Import ssreflect ssrfun ssrbool eqtype ssrnat choice seq",
"From mathcomp Require Import div fintype tuple finfun bigop fingroup perm",
"From mathcomp Require Import ssralg zmodp matrix mxalgebra poly polydiv"
] | algebra/mxpoly.v | Cayley_Hamilton | |
eigenvalue_root_char(F : fieldType) n (A : 'M[F]_n) a :
eigenvalue A a = root (char_poly A) a.
Proof.
transitivity (\det (a%:M - A) == 0).
apply/eigenvalueP/det0P=> [[v Av_av v_nz] | [v v_nz Av_av]]; exists v => //.
by rewrite mulmxBr Av_av mul_mx_scalar subrr.
by apply/eqP; rewrite -mul_mx_scalar eq_sym -sub... | Lemma | algebra | [
"From HB Require Import structures",
"From mathcomp Require Import ssreflect ssrfun ssrbool eqtype ssrnat choice seq",
"From mathcomp Require Import div fintype tuple finfun bigop fingroup perm",
"From mathcomp Require Import ssralg zmodp matrix mxalgebra poly polydiv"
] | algebra/mxpoly.v | eigenvalue_root_char | |
char_poly_trig{R : comNzRingType} n (A : 'M[R]_n) : is_trig_mx A ->
char_poly A = \prod_(i < n) ('X - (A i i)%:P).
Proof.
move=> /is_trig_mxP Atrig; rewrite /char_poly det_trig.
by apply: eq_bigr => i; rewrite !mxE eqxx.
by apply/is_trig_mxP => i j lt_ij; rewrite !mxE -val_eqE ltn_eqF ?Atrig ?subrr.
Qed. | Lemma | algebra | [
"From HB Require Import structures",
"From mathcomp Require Import ssreflect ssrfun ssrbool eqtype ssrnat choice seq",
"From mathcomp Require Import div fintype tuple finfun bigop fingroup perm",
"From mathcomp Require Import ssralg zmodp matrix mxalgebra poly polydiv"
] | algebra/mxpoly.v | char_poly_trig | |
companionmx{R : nzRingType} (p : seq R) (d := (size p).-1) :=
\matrix_(i < d, j < d)
if (i == d.-1 :> nat) then - p`_j else (i.+1 == j :> nat)%:R. | Definition | algebra | [
"From HB Require Import structures",
"From mathcomp Require Import ssreflect ssrfun ssrbool eqtype ssrnat choice seq",
"From mathcomp Require Import div fintype tuple finfun bigop fingroup perm",
"From mathcomp Require Import ssralg zmodp matrix mxalgebra poly polydiv"
] | algebra/mxpoly.v | companionmx | |
companionmxK{R : comNzRingType} (p : {poly R}) :
p \is monic -> char_poly (companionmx p) = p.
Proof.
pose D n : 'M[{poly R}]_n := \matrix_(i, j)
('X *+ (i == j.+1 :> nat) - ((i == j)%:R)%:P).
have detD n : \det (D n) = (-1) ^+ n.
elim: n => [|n IHn]; first by rewrite det_mx00.
rewrite (expand_det_row _ ord0)... | Lemma | algebra | [
"From HB Require Import structures",
"From mathcomp Require Import ssreflect ssrfun ssrbool eqtype ssrnat choice seq",
"From mathcomp Require Import div fintype tuple finfun bigop fingroup perm",
"From mathcomp Require Import ssralg zmodp matrix mxalgebra poly polydiv"
] | algebra/mxpoly.v | companionmxK | |
mulmx_delta_companion(R : nzRingType) (p : seq R)
(i: 'I_(size p).-1) (i_small : i.+1 < (size p).-1):
delta_mx 0 i *m companionmx p = delta_mx 0 (Ordinal i_small) :> 'rV__.
Proof.
apply/rowP => j; rewrite !mxE (bigD1 i) //= ?(=^~val_eqE, mxE) /= eqxx mul1r.
rewrite ltn_eqF ?big1 ?addr0 1?eq_sym //; last first.
by... | Lemma | algebra | [
"From HB Require Import structures",
"From mathcomp Require Import ssreflect ssrfun ssrbool eqtype ssrnat choice seq",
"From mathcomp Require Import div fintype tuple finfun bigop fingroup perm",
"From mathcomp Require Import ssralg zmodp matrix mxalgebra poly polydiv"
] | algebra/mxpoly.v | mulmx_delta_companion | |
row'_col'_char_poly_mx{R : nzRingType} m i (M : 'M[R]_m) :
row' i (col' i (char_poly_mx M)) = char_poly_mx (row' i (col' i M)).
Proof. by apply/matrixP => k l; rewrite !mxE (inj_eq lift_inj). Qed. | Lemma | algebra | [
"From HB Require Import structures",
"From mathcomp Require Import ssreflect ssrfun ssrbool eqtype ssrnat choice seq",
"From mathcomp Require Import div fintype tuple finfun bigop fingroup perm",
"From mathcomp Require Import ssralg zmodp matrix mxalgebra poly polydiv"
] | algebra/mxpoly.v | row'_col'_char_poly_mx | |
char_block_diag_mx{R : nzRingType} m n (A : 'M[R]_m) (B : 'M[R]_n) :
char_poly_mx (block_mx A 0 0 B) =
block_mx (char_poly_mx A) 0 0 (char_poly_mx B).
Proof.
rewrite /char_poly_mx map_block_mx/= !map_mx0.
by rewrite scalar_mx_block opp_block_mx add_block_mx !subr0.
Qed. | Lemma | algebra | [
"From HB Require Import structures",
"From mathcomp Require Import ssreflect ssrfun ssrbool eqtype ssrnat choice seq",
"From mathcomp Require Import div fintype tuple finfun bigop fingroup perm",
"From mathcomp Require Import ssralg zmodp matrix mxalgebra poly polydiv"
] | algebra/mxpoly.v | char_block_diag_mx | |
degree_mxminpoly:= ex_minn degree_mxminpoly_proof.
Local Notation d := degree_mxminpoly.
Local Notation Ad := (powers_mx A d). | Definition | algebra | [
"From HB Require Import structures",
"From mathcomp Require Import ssreflect ssrfun ssrbool eqtype ssrnat choice seq",
"From mathcomp Require Import div fintype tuple finfun bigop fingroup perm",
"From mathcomp Require Import ssralg zmodp matrix mxalgebra poly polydiv"
] | algebra/mxpoly.v | degree_mxminpoly | |
mxminpoly_nonconstant: d > 0.
Proof.
rewrite /d; case: ex_minnP => -[] //; rewrite leqn0 mxrank_eq0; move/eqP.
by move/row_matrixP/(_ 0)/eqP; rewrite rowK row0 mxvec_eq0 -mxrank_eq0 mxrank1.
Qed. | Lemma | algebra | [
"From HB Require Import structures",
"From mathcomp Require Import ssreflect ssrfun ssrbool eqtype ssrnat choice seq",
"From mathcomp Require Import div fintype tuple finfun bigop fingroup perm",
"From mathcomp Require Import ssralg zmodp matrix mxalgebra poly polydiv"
] | algebra/mxpoly.v | mxminpoly_nonconstant | |
minpoly_mx1: (1%:M \in Ad)%MS.
Proof.
by apply: (eq_row_sub (Ordinal mxminpoly_nonconstant)); rewrite rowK.
Qed. | Lemma | algebra | [
"From HB Require Import structures",
"From mathcomp Require Import ssreflect ssrfun ssrbool eqtype ssrnat choice seq",
"From mathcomp Require Import div fintype tuple finfun bigop fingroup perm",
"From mathcomp Require Import ssralg zmodp matrix mxalgebra poly polydiv"
] | algebra/mxpoly.v | minpoly_mx1 | |
minpoly_mx_free: row_free Ad.
Proof.
have:= mxminpoly_nonconstant; rewrite /d; case: ex_minnP => -[] // d' _ /(_ d').
by move/implyP; rewrite ltnn implybF -ltnS ltn_neqAle rank_leq_row andbT negbK.
Qed. | Lemma | algebra | [
"From HB Require Import structures",
"From mathcomp Require Import ssreflect ssrfun ssrbool eqtype ssrnat choice seq",
"From mathcomp Require Import div fintype tuple finfun bigop fingroup perm",
"From mathcomp Require Import ssralg zmodp matrix mxalgebra poly polydiv"
] | algebra/mxpoly.v | minpoly_mx_free | |
horner_mx_memp : (horner_mx A p \in Ad)%MS.
Proof.
elim/poly_ind: p => [|p a IHp]; first by rewrite rmorph0 // linear0 sub0mx.
rewrite rmorphD rmorphM /= horner_mx_C horner_mx_X.
rewrite addrC -scalemx1 linearP /= -(mul_vec_lin (mulmxr A)).
case/submxP: IHp => u ->{p}.
have: (powers_mx A (1 + d) <= Ad)%MS.
rewrite -(... | Lemma | algebra | [
"From HB Require Import structures",
"From mathcomp Require Import ssreflect ssrfun ssrbool eqtype ssrnat choice seq",
"From mathcomp Require Import div fintype tuple finfun bigop fingroup perm",
"From mathcomp Require Import ssralg zmodp matrix mxalgebra poly polydiv"
] | algebra/mxpoly.v | horner_mx_mem | |
mx_inv_hornerB := rVpoly (mxvec B *m pinvmx Ad). | Definition | algebra | [
"From HB Require Import structures",
"From mathcomp Require Import ssreflect ssrfun ssrbool eqtype ssrnat choice seq",
"From mathcomp Require Import div fintype tuple finfun bigop fingroup perm",
"From mathcomp Require Import ssralg zmodp matrix mxalgebra poly polydiv"
] | algebra/mxpoly.v | mx_inv_horner | |
mx_inv_horner0: mx_inv_horner 0 = 0.
Proof. by rewrite /mx_inv_horner !(linear0, mul0mx). Qed. | Lemma | algebra | [
"From HB Require Import structures",
"From mathcomp Require Import ssreflect ssrfun ssrbool eqtype ssrnat choice seq",
"From mathcomp Require Import div fintype tuple finfun bigop fingroup perm",
"From mathcomp Require Import ssralg zmodp matrix mxalgebra poly polydiv"
] | algebra/mxpoly.v | mx_inv_horner0 | |
mx_inv_hornerKB : (B \in Ad)%MS -> horner_mx A (mx_inv_horner B) = B.
Proof. by move=> sBAd; rewrite horner_rVpoly mulmxKpV ?mxvecK. Qed. | Lemma | algebra | [
"From HB Require Import structures",
"From mathcomp Require Import ssreflect ssrfun ssrbool eqtype ssrnat choice seq",
"From mathcomp Require Import div fintype tuple finfun bigop fingroup perm",
"From mathcomp Require Import ssralg zmodp matrix mxalgebra poly polydiv"
] | algebra/mxpoly.v | mx_inv_hornerK | |
minpoly_mxMB C : (B \in Ad -> C \in Ad -> B * C \in Ad)%MS.
Proof.
move=> AdB AdC; rewrite -(mx_inv_hornerK AdB) -(mx_inv_hornerK AdC).
by rewrite -rmorphM ?horner_mx_mem.
Qed. | Lemma | algebra | [
"From HB Require Import structures",
"From mathcomp Require Import ssreflect ssrfun ssrbool eqtype ssrnat choice seq",
"From mathcomp Require Import div fintype tuple finfun bigop fingroup perm",
"From mathcomp Require Import ssralg zmodp matrix mxalgebra poly polydiv"
] | algebra/mxpoly.v | minpoly_mxM | |
minpoly_mx_ring: mxring Ad.
Proof.
apply/andP; split; first exact/mulsmx_subP/minpoly_mxM.
apply/mxring_idP; exists 1%:M; split=> *; rewrite ?mulmx1 ?mul1mx //.
by rewrite -mxrank_eq0 mxrank1.
exact: minpoly_mx1.
Qed. | Lemma | algebra | [
"From HB Require Import structures",
"From mathcomp Require Import ssreflect ssrfun ssrbool eqtype ssrnat choice seq",
"From mathcomp Require Import div fintype tuple finfun bigop fingroup perm",
"From mathcomp Require Import ssralg zmodp matrix mxalgebra poly polydiv"
] | algebra/mxpoly.v | minpoly_mx_ring | |
mxminpoly:= 'X^d - mx_inv_horner (A ^+ d).
Local Notation p_A := mxminpoly. | Definition | algebra | [
"From HB Require Import structures",
"From mathcomp Require Import ssreflect ssrfun ssrbool eqtype ssrnat choice seq",
"From mathcomp Require Import div fintype tuple finfun bigop fingroup perm",
"From mathcomp Require Import ssralg zmodp matrix mxalgebra poly polydiv"
] | algebra/mxpoly.v | mxminpoly | |
size_mxminpoly: size p_A = d.+1.
Proof. by rewrite size_polyDl ?size_polyXn // size_polyN ltnS size_poly. Qed. | Lemma | algebra | [
"From HB Require Import structures",
"From mathcomp Require Import ssreflect ssrfun ssrbool eqtype ssrnat choice seq",
"From mathcomp Require Import div fintype tuple finfun bigop fingroup perm",
"From mathcomp Require Import ssralg zmodp matrix mxalgebra poly polydiv"
] | algebra/mxpoly.v | size_mxminpoly | |
mxminpoly_monic: p_A \is monic.
Proof.
rewrite monicE /lead_coef size_mxminpoly coefB coefXn eqxx /=.
by rewrite nth_default ?size_poly // subr0.
Qed. | Lemma | algebra | [
"From HB Require Import structures",
"From mathcomp Require Import ssreflect ssrfun ssrbool eqtype ssrnat choice seq",
"From mathcomp Require Import div fintype tuple finfun bigop fingroup perm",
"From mathcomp Require Import ssralg zmodp matrix mxalgebra poly polydiv"
] | algebra/mxpoly.v | mxminpoly_monic | |
size_mod_mxminpolyp : size (p %% p_A) <= d.
Proof.
by rewrite -ltnS -size_mxminpoly ltn_modp // -size_poly_eq0 size_mxminpoly.
Qed. | Lemma | algebra | [
"From HB Require Import structures",
"From mathcomp Require Import ssreflect ssrfun ssrbool eqtype ssrnat choice seq",
"From mathcomp Require Import div fintype tuple finfun bigop fingroup perm",
"From mathcomp Require Import ssralg zmodp matrix mxalgebra poly polydiv"
] | algebra/mxpoly.v | size_mod_mxminpoly | |
mx_root_minpoly: horner_mx A p_A = 0.
Proof.
rewrite rmorphB -{3}(horner_mx_X A) -rmorphXn /=.
by rewrite mx_inv_hornerK ?subrr ?horner_mx_mem.
Qed. | Lemma | algebra | [
"From HB Require Import structures",
"From mathcomp Require Import ssreflect ssrfun ssrbool eqtype ssrnat choice seq",
"From mathcomp Require Import div fintype tuple finfun bigop fingroup perm",
"From mathcomp Require Import ssralg zmodp matrix mxalgebra poly polydiv"
] | algebra/mxpoly.v | mx_root_minpoly | |
horner_rVpolyK(u : 'rV_d) :
mx_inv_horner (horner_mx A (rVpoly u)) = rVpoly u.
Proof.
congr rVpoly; rewrite horner_rVpoly vec_mxK.
by apply: (row_free_inj minpoly_mx_free); rewrite mulmxKpV ?submxMl.
Qed. | Lemma | algebra | [
"From HB Require Import structures",
"From mathcomp Require Import ssreflect ssrfun ssrbool eqtype ssrnat choice seq",
"From mathcomp Require Import div fintype tuple finfun bigop fingroup perm",
"From mathcomp Require Import ssralg zmodp matrix mxalgebra poly polydiv"
] | algebra/mxpoly.v | horner_rVpolyK | |
horner_mxKp : mx_inv_horner (horner_mx A p) = p %% p_A.
Proof.
rewrite {1}(Pdiv.IdomainMonic.divp_eq mxminpoly_monic p) rmorphD rmorphM /=.
rewrite mx_root_minpoly mulr0 add0r.
by rewrite -(poly_rV_K (size_mod_mxminpoly _)) horner_rVpolyK.
Qed. | Lemma | algebra | [
"From HB Require Import structures",
"From mathcomp Require Import ssreflect ssrfun ssrbool eqtype ssrnat choice seq",
"From mathcomp Require Import div fintype tuple finfun bigop fingroup perm",
"From mathcomp Require Import ssralg zmodp matrix mxalgebra poly polydiv"
] | algebra/mxpoly.v | horner_mxK | |
mxminpoly_minp : horner_mx A p = 0 -> p_A %| p.
Proof. by move=> pA0; rewrite /dvdp -horner_mxK pA0 mx_inv_horner0. Qed. | Lemma | algebra | [
"From HB Require Import structures",
"From mathcomp Require Import ssreflect ssrfun ssrbool eqtype ssrnat choice seq",
"From mathcomp Require Import div fintype tuple finfun bigop fingroup perm",
"From mathcomp Require Import ssralg zmodp matrix mxalgebra poly polydiv"
] | algebra/mxpoly.v | mxminpoly_min | |
mxminpoly_minPp : reflect (horner_mx A p = 0) (p_A %| p).
Proof.
apply: (iffP idP); last exact: mxminpoly_min.
by move=> /Pdiv.Field.dvdpP[q ->]; rewrite rmorphM/= mx_root_minpoly mulr0.
Qed. | Lemma | algebra | [
"From HB Require Import structures",
"From mathcomp Require Import ssreflect ssrfun ssrbool eqtype ssrnat choice seq",
"From mathcomp Require Import div fintype tuple finfun bigop fingroup perm",
"From mathcomp Require Import ssralg zmodp matrix mxalgebra poly polydiv"
] | algebra/mxpoly.v | mxminpoly_minP | |
dvd_mxminpolyp : (p_A %| p) = (horner_mx A p == 0).
Proof. exact/mxminpoly_minP/eqP. Qed. | Lemma | algebra | [
"From HB Require Import structures",
"From mathcomp Require Import ssreflect ssrfun ssrbool eqtype ssrnat choice seq",
"From mathcomp Require Import div fintype tuple finfun bigop fingroup perm",
"From mathcomp Require Import ssralg zmodp matrix mxalgebra poly polydiv"
] | algebra/mxpoly.v | dvd_mxminpoly | |
horner_rVpoly_inj: injective (horner_mx A \o rVpoly : 'rV_d -> 'M_n).
Proof.
apply: can_inj (poly_rV \o mx_inv_horner) _ => u /=.
by rewrite horner_rVpolyK rVpolyK.
Qed. | Lemma | algebra | [
"From HB Require Import structures",
"From mathcomp Require Import ssreflect ssrfun ssrbool eqtype ssrnat choice seq",
"From mathcomp Require Import div fintype tuple finfun bigop fingroup perm",
"From mathcomp Require Import ssralg zmodp matrix mxalgebra poly polydiv"
] | algebra/mxpoly.v | horner_rVpoly_inj | |
mxminpoly_linear_is_scalar: (d <= 1) = is_scalar_mx A.
Proof.
have scalP := has_non_scalar_mxP minpoly_mx1.
rewrite leqNgt -(eqnP minpoly_mx_free); apply/scalP/idP=> [|[[B]]].
case scalA: (is_scalar_mx A); [by right | left].
by exists A; rewrite ?scalA // -{1}(horner_mx_X A) horner_mx_mem.
move/mx_inv_hornerK=> <- ... | Lemma | algebra | [
"From HB Require Import structures",
"From mathcomp Require Import ssreflect ssrfun ssrbool eqtype ssrnat choice seq",
"From mathcomp Require Import div fintype tuple finfun bigop fingroup perm",
"From mathcomp Require Import ssralg zmodp matrix mxalgebra poly polydiv"
] | algebra/mxpoly.v | mxminpoly_linear_is_scalar | |
mxminpoly_dvd_char: p_A %| char_poly A.
Proof. exact/mxminpoly_min/Cayley_Hamilton. Qed. | Lemma | algebra | [
"From HB Require Import structures",
"From mathcomp Require Import ssreflect ssrfun ssrbool eqtype ssrnat choice seq",
"From mathcomp Require Import div fintype tuple finfun bigop fingroup perm",
"From mathcomp Require Import ssralg zmodp matrix mxalgebra poly polydiv"
] | algebra/mxpoly.v | mxminpoly_dvd_char | |
eigenvalue_root_mina : eigenvalue A a = root p_A a.
Proof.
apply/idP/idP=> Aa; last first.
rewrite eigenvalue_root_char !root_factor_theorem in Aa *.
exact: dvdp_trans Aa mxminpoly_dvd_char.
have{Aa} [v Av_av v_nz] := eigenvalueP Aa.
apply: contraR v_nz => pa_nz; rewrite -{pa_nz}(eqmx_eq0 (eqmx_scale _ pa_nz)).
app... | Lemma | algebra | [
"From HB Require Import structures",
"From mathcomp Require Import ssreflect ssrfun ssrbool eqtype ssrnat choice seq",
"From mathcomp Require Import div fintype tuple finfun bigop fingroup perm",
"From mathcomp Require Import ssralg zmodp matrix mxalgebra poly polydiv"
] | algebra/mxpoly.v | eigenvalue_root_min |
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