statement stringlengths 1 4.33k | proof stringlengths 0 37.9k | type stringclasses 25
values | symbolic_name stringlengths 1 67 | library stringclasses 10
values | filename stringclasses 112
values | imports listlengths 2 138 | deps listlengths 0 64 | docstring stringclasses 798
values | source_url stringclasses 1
value | commit stringclasses 1
value |
|---|---|---|---|---|---|---|---|---|---|---|
Remx_rect : {in realmx &, forall A B, (A + 'i *: B) ^ (@Re _) = A}. | Proof.
move=> A B Areal Breal; apply/matrixP=> i j; rewrite !mxE.
by rewrite Re_rect // (mxOverP _ _).
Qed. | Lemma | Remx_rect | algebra | algebra/spectral.v | [
"HB",
"structures",
"mathcomp",
"ssreflect",
"ssrfun",
"ssrbool",
"eqtype",
"ssrnat",
"seq",
"choice",
"fintype",
"bigop",
"nmodule",
"order",
"rings_modules_and_algebras",
"divalg",
"poly",
"matrix",
"mxalgebra",
"mxpoly",
"mxred",
"orderedzmod",
"numdomain",
"numfield... | [
"Re",
"Re_rect",
"apply",
"matrixP",
"mxE",
"mxOverP",
"realmx"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
Immx_rect : {in realmx &, forall A B, (A + 'i *: B) ^ (@Im _) = B}. | Proof.
move=> /= A B Areal Breal; apply/matrixP=> i j; rewrite !mxE.
by rewrite Im_rect // (mxOverP _ _).
Qed. | Lemma | Immx_rect | algebra | algebra/spectral.v | [
"HB",
"structures",
"mathcomp",
"ssreflect",
"ssrfun",
"ssrbool",
"eqtype",
"ssrnat",
"seq",
"choice",
"fintype",
"bigop",
"nmodule",
"order",
"rings_modules_and_algebras",
"divalg",
"poly",
"matrix",
"mxalgebra",
"mxpoly",
"mxred",
"orderedzmod",
"numdomain",
"numfield... | [
"Im",
"Im_rect",
"apply",
"matrixP",
"mxE",
"mxOverP",
"realmx"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
eqmx_ReiIm A B A' B' :
A \is a realmx -> B \is a realmx -> A' \is a realmx -> B' \is a realmx ->
(A + 'i *: B) = (A' + 'i *: B') -> (A, B) = (A', B'). | Proof.
move=> ARe BRe A'Im B'Im eqAB.
have /(congr1 (fun A => A ^ (@Im _))) := eqAB.
have /(congr1 (fun A => A ^ (@Re _))) := eqAB.
by rewrite !Remx_rect// !Immx_rect// => -> ->.
Qed. | Lemma | eqmx_ReiIm | algebra | algebra/spectral.v | [
"HB",
"structures",
"mathcomp",
"ssreflect",
"ssrfun",
"ssrbool",
"eqtype",
"ssrnat",
"seq",
"choice",
"fintype",
"bigop",
"nmodule",
"order",
"rings_modules_and_algebras",
"divalg",
"poly",
"matrix",
"mxalgebra",
"mxpoly",
"mxred",
"orderedzmod",
"numdomain",
"numfield... | [
"A'",
"Im",
"Immx_rect",
"Re",
"Remx_rect",
"realmx"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
realsym_hermsym {C : numClosedFieldType} {n} (A : 'M[C]_n) :
A \is symmetricmx -> A \is a realmx -> A \is hermsymmx. | Proof.
move=> Asym Areal; apply/is_hermitianmxP.
by rewrite (trmx_hermitian (HermitianMx Asym))/= !scale1r ?realmxC ?map_mx_id.
Qed. | Lemma | realsym_hermsym | algebra | algebra/spectral.v | [
"HB",
"structures",
"mathcomp",
"ssreflect",
"ssrfun",
"ssrbool",
"eqtype",
"ssrnat",
"seq",
"choice",
"fintype",
"bigop",
"nmodule",
"order",
"rings_modules_and_algebras",
"divalg",
"poly",
"matrix",
"mxalgebra",
"mxpoly",
"mxred",
"orderedzmod",
"numdomain",
"numfield... | [
"apply",
"hermsymmx",
"is_hermitianmxP",
"map_mx_id",
"realmx",
"realmxC",
"scale1r",
"symmetricmx",
"trmx_hermitian"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
real_similar {C : numClosedFieldType} {n} (A B : 'M[C]_n) :
similar_in unitmx A B ->
A \is a realmx -> B \is a realmx -> similar_in [predI realmx & unitmx] A B. | Proof.
case=> [P /=]; pose Pr := P ^ (@Re _); pose Pi := P ^ (@Im _).
have Pr_real : Pr \is a realmx by apply/mxOverP=> i j; rewrite !mxE Creal_Re.
have Pi_real : Pi \is a realmx by apply/mxOverP=> i j; rewrite !mxE Creal_Im.
pose Q x := P ^ (@Re _) + x *: P ^ (@Im _).
have -> : P = Q 'i by apply/matrixP=> i j; rewrite... | Lemma | real_similar | algebra | algebra/spectral.v | [
"HB",
"structures",
"mathcomp",
"ssreflect",
"ssrfun",
"ssrbool",
"eqtype",
"ssrnat",
"seq",
"choice",
"fintype",
"bigop",
"nmodule",
"order",
"rings_modules_and_algebras",
"divalg",
"poly",
"matrix",
"mxalgebra",
"mxpoly",
"mxred",
"orderedzmod",
"numdomain",
"numfield... | [
"Creal_Im",
"Creal_Re",
"Crect",
"Im",
"Re",
"allPn",
"apply",
"contra_neq",
"det_map_mx",
"eqmx_ReiIm",
"hornerC",
"hornerE",
"hornerX",
"horner_eval",
"horner_evalE",
"id",
"inE",
"iota",
"iota_uniq",
"ltnn",
"mapP",
"map_inj_uniq",
"map_mxD",
"map_mxZ",
"map_mx_com... | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
unitarymx {m n} | := [qualify X : 'M[C]_(m, n) | X *m X ^t* == 1%:M]. | Definition | unitarymx | algebra | algebra/spectral.v | [
"HB",
"structures",
"mathcomp",
"ssreflect",
"ssrfun",
"ssrbool",
"eqtype",
"ssrnat",
"seq",
"choice",
"fintype",
"bigop",
"nmodule",
"order",
"rings_modules_and_algebras",
"divalg",
"poly",
"matrix",
"mxalgebra",
"mxpoly",
"mxred",
"orderedzmod",
"numdomain",
"numfield... | [] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
unitarymx_key m n : pred_key (@unitarymx m n). | Proof. by []. Qed. | Fact | unitarymx_key | algebra | algebra/spectral.v | [
"HB",
"structures",
"mathcomp",
"ssreflect",
"ssrfun",
"ssrbool",
"eqtype",
"ssrnat",
"seq",
"choice",
"fintype",
"bigop",
"nmodule",
"order",
"rings_modules_and_algebras",
"divalg",
"poly",
"matrix",
"mxalgebra",
"mxpoly",
"mxred",
"orderedzmod",
"numdomain",
"numfield... | [
"unitarymx"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
unitarymx_keyed m n | := KeyedQualifier (unitarymx_key m n). | Canonical | unitarymx_keyed | algebra | algebra/spectral.v | [
"HB",
"structures",
"mathcomp",
"ssreflect",
"ssrfun",
"ssrbool",
"eqtype",
"ssrnat",
"seq",
"choice",
"fintype",
"bigop",
"nmodule",
"order",
"rings_modules_and_algebras",
"divalg",
"poly",
"matrix",
"mxalgebra",
"mxpoly",
"mxred",
"orderedzmod",
"numdomain",
"numfield... | [
"unitarymx_key"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
unitarymxP m n {M : 'M[C]_(m, n)} :
reflect (M *m M^t* = 1%:M) (M \is unitarymx). | Proof. by apply: (iffP eqP). Qed. | Lemma | unitarymxP | algebra | algebra/spectral.v | [
"HB",
"structures",
"mathcomp",
"ssreflect",
"ssrfun",
"ssrbool",
"eqtype",
"ssrnat",
"seq",
"choice",
"fintype",
"bigop",
"nmodule",
"order",
"rings_modules_and_algebras",
"divalg",
"poly",
"matrix",
"mxalgebra",
"mxpoly",
"mxred",
"orderedzmod",
"numdomain",
"numfield... | [
"apply",
"unitarymx"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
mulmxtVK m1 m2 n (A : 'M[C]_(m1, n)) (B : 'M[C]_(n, m2)) :
B \is unitarymx -> A *m B *m B^t* = A. | Proof. by move=> B_unitary; rewrite -mulmxA (unitarymxP _) ?mulmx1. Qed. | Lemma | mulmxtVK | algebra | algebra/spectral.v | [
"HB",
"structures",
"mathcomp",
"ssreflect",
"ssrfun",
"ssrbool",
"eqtype",
"ssrnat",
"seq",
"choice",
"fintype",
"bigop",
"nmodule",
"order",
"rings_modules_and_algebras",
"divalg",
"poly",
"matrix",
"mxalgebra",
"mxpoly",
"mxred",
"orderedzmod",
"numdomain",
"numfield... | [
"mulmx1",
"mulmxA",
"unitarymx",
"unitarymxP"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
unitarymx_unit n (M : 'M[C]_n) : M \is unitarymx -> M \in unitmx. | Proof. by move=> /unitarymxP /mulmx1_unit []. Qed. | Lemma | unitarymx_unit | algebra | algebra/spectral.v | [
"HB",
"structures",
"mathcomp",
"ssreflect",
"ssrfun",
"ssrbool",
"eqtype",
"ssrnat",
"seq",
"choice",
"fintype",
"bigop",
"nmodule",
"order",
"rings_modules_and_algebras",
"divalg",
"poly",
"matrix",
"mxalgebra",
"mxpoly",
"mxred",
"orderedzmod",
"numdomain",
"numfield... | [
"mulmx1_unit",
"unitarymx",
"unitarymxP",
"unitmx"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
invmx_unitary n (M : 'M[C]_n) : M \is unitarymx -> invmx M = M^t*. | Proof.
move=> Munitary; apply: (@row_full_inj _ _ _ _ M).
by rewrite row_full_unit unitarymx_unit.
by rewrite mulmxV ?unitarymx_unit ?(unitarymxP _).
Qed. | Lemma | invmx_unitary | algebra | algebra/spectral.v | [
"HB",
"structures",
"mathcomp",
"ssreflect",
"ssrfun",
"ssrbool",
"eqtype",
"ssrnat",
"seq",
"choice",
"fintype",
"bigop",
"nmodule",
"order",
"rings_modules_and_algebras",
"divalg",
"poly",
"matrix",
"mxalgebra",
"mxpoly",
"mxred",
"orderedzmod",
"numdomain",
"numfield... | [
"apply",
"invmx",
"mulmxV",
"row_full_inj",
"row_full_unit",
"unitarymx",
"unitarymxP",
"unitarymx_unit"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
mulmxKtV m1 m2 n (A : 'M[C]_(m1, n)) (B : 'M[C]_(m2, n)) :
B \is unitarymx -> m2 = n -> A *m B^t* *m B = A. | Proof.
move=> B_unitary m2E; case: _ / (esym m2E) in B B_unitary *.
by rewrite -invmx_unitary // mulmxKV //; exact: unitarymx_unit.
Qed. | Lemma | mulmxKtV | algebra | algebra/spectral.v | [
"HB",
"structures",
"mathcomp",
"ssreflect",
"ssrfun",
"ssrbool",
"eqtype",
"ssrnat",
"seq",
"choice",
"fintype",
"bigop",
"nmodule",
"order",
"rings_modules_and_algebras",
"divalg",
"poly",
"matrix",
"mxalgebra",
"mxpoly",
"mxred",
"orderedzmod",
"numdomain",
"numfield... | [
"invmx_unitary",
"mulmxKV",
"unitarymx",
"unitarymx_unit"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
mxrank_unitary m n (M : 'M[C]_(m, n)) : M \is unitarymx -> \rank M = m. | Proof.
rewrite qualifE => /eqP /(congr1 mxrank); rewrite mxrank1 => rkM.
apply/eqP; rewrite eqn_leq rank_leq_row /= -[X in (X <= _)%N]rkM.
by rewrite mxrankM_maxl.
Qed. | Lemma | mxrank_unitary | algebra | algebra/spectral.v | [
"HB",
"structures",
"mathcomp",
"ssreflect",
"ssrfun",
"ssrbool",
"eqtype",
"ssrnat",
"seq",
"choice",
"fintype",
"bigop",
"nmodule",
"order",
"rings_modules_and_algebras",
"divalg",
"poly",
"matrix",
"mxalgebra",
"mxpoly",
"mxred",
"orderedzmod",
"numdomain",
"numfield... | [
"apply",
"eqn_leq",
"mxrank",
"mxrank1",
"mxrankM_maxl",
"rank",
"rank_leq_row",
"unitarymx"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
mul_unitarymx m n p (A : 'M[C]_(m, n)) (B : 'M[C]_(n, p)) :
A \is unitarymx -> B \is unitarymx -> A *m B \is unitarymx. | Proof.
move=> Aunitary Bunitary; apply/unitarymxP; rewrite trmx_mul map_mxM.
by rewrite mulmxA -[A *m _ *m _]mulmxA !(unitarymxP _, mulmx1).
Qed. | Lemma | mul_unitarymx | algebra | algebra/spectral.v | [
"HB",
"structures",
"mathcomp",
"ssreflect",
"ssrfun",
"ssrbool",
"eqtype",
"ssrnat",
"seq",
"choice",
"fintype",
"bigop",
"nmodule",
"order",
"rings_modules_and_algebras",
"divalg",
"poly",
"matrix",
"mxalgebra",
"mxpoly",
"mxred",
"orderedzmod",
"numdomain",
"numfield... | [
"apply",
"map_mxM",
"mulmx1",
"mulmxA",
"trmx_mul",
"unitarymx",
"unitarymxP"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
pinvmx_unitary n (M : 'M[C]_n) : M \is unitarymx -> pinvmx M = M^t*. | Proof. by move=> Munitary; rewrite pinvmxE ?unitarymx_unit// invmx_unitary. Qed. | Lemma | pinvmx_unitary | algebra | algebra/spectral.v | [
"HB",
"structures",
"mathcomp",
"ssreflect",
"ssrfun",
"ssrbool",
"eqtype",
"ssrnat",
"seq",
"choice",
"fintype",
"bigop",
"nmodule",
"order",
"rings_modules_and_algebras",
"divalg",
"poly",
"matrix",
"mxalgebra",
"mxpoly",
"mxred",
"orderedzmod",
"numdomain",
"numfield... | [
"invmx_unitary",
"pinvmx",
"pinvmxE",
"unitarymx",
"unitarymx_unit"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
conjymx n (P M : 'M[C]_n) : P \is unitarymx -> conjmx P M = P *m M *m P^t*. | Proof. by move=> Munitary; rewrite conjumx ?invmx_unitary ?unitarymx_unit. Qed. | Lemma | conjymx | algebra | algebra/spectral.v | [
"HB",
"structures",
"mathcomp",
"ssreflect",
"ssrfun",
"ssrbool",
"eqtype",
"ssrnat",
"seq",
"choice",
"fintype",
"bigop",
"nmodule",
"order",
"rings_modules_and_algebras",
"divalg",
"poly",
"matrix",
"mxalgebra",
"mxpoly",
"mxred",
"orderedzmod",
"numdomain",
"numfield... | [
"conjmx",
"conjumx",
"invmx_unitary",
"unitarymx",
"unitarymx_unit"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
trmx_unitary n (M : 'M[C]_n) : (M ^T \is unitarymx) = (M \is unitarymx). | Proof.
apply/unitarymxP/unitarymxP; rewrite -?map_trmx -trmx_mul.
by rewrite -trmx1 => /trmx_inj /mulmx1C->; rewrite trmx1.
by move=> /mulmx1C->; rewrite trmx1.
Qed. | Lemma | trmx_unitary | algebra | algebra/spectral.v | [
"HB",
"structures",
"mathcomp",
"ssreflect",
"ssrfun",
"ssrbool",
"eqtype",
"ssrnat",
"seq",
"choice",
"fintype",
"bigop",
"nmodule",
"order",
"rings_modules_and_algebras",
"divalg",
"poly",
"matrix",
"mxalgebra",
"mxpoly",
"mxred",
"orderedzmod",
"numdomain",
"numfield... | [
"apply",
"map_trmx",
"mulmx1C",
"trmx1",
"trmx_inj",
"trmx_mul",
"unitarymx",
"unitarymxP"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
conjC_unitary m n (M : 'M[C]_(m, n)) :
(M ^ conjC \is unitarymx) = (M \is unitarymx). | Proof.
apply/unitarymxP/unitarymxP; rewrite -?map_mxM ?map_trmx; last first.
by move=> ->; rewrite map_mx1.
by rewrite -[1%:M](map_mx1 conjC) => /map_mx_inj ->; rewrite map_mx1.
Qed. | Lemma | conjC_unitary | algebra | algebra/spectral.v | [
"HB",
"structures",
"mathcomp",
"ssreflect",
"ssrfun",
"ssrbool",
"eqtype",
"ssrnat",
"seq",
"choice",
"fintype",
"bigop",
"nmodule",
"order",
"rings_modules_and_algebras",
"divalg",
"poly",
"matrix",
"mxalgebra",
"mxpoly",
"mxred",
"orderedzmod",
"numdomain",
"numfield... | [
"apply",
"conjC",
"last",
"map_mx1",
"map_mxM",
"map_mx_inj",
"map_trmx",
"unitarymx",
"unitarymxP"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
trmxC_unitary n (M : 'M[C]_n) : (M ^t* \is unitarymx) = (M \is unitarymx). | Proof. by rewrite conjC_unitary trmx_unitary. Qed. | Lemma | trmxC_unitary | algebra | algebra/spectral.v | [
"HB",
"structures",
"mathcomp",
"ssreflect",
"ssrfun",
"ssrbool",
"eqtype",
"ssrnat",
"seq",
"choice",
"fintype",
"bigop",
"nmodule",
"order",
"rings_modules_and_algebras",
"divalg",
"poly",
"matrix",
"mxalgebra",
"mxpoly",
"mxred",
"orderedzmod",
"numdomain",
"numfield... | [
"conjC_unitary",
"trmx_unitary",
"unitarymx"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
normalmx | := [qualify M : 'M[C]_n | M *m M ^t* == M ^t* *m M]. | Definition | normalmx | algebra | algebra/spectral.v | [
"HB",
"structures",
"mathcomp",
"ssreflect",
"ssrfun",
"ssrbool",
"eqtype",
"ssrnat",
"seq",
"choice",
"fintype",
"bigop",
"nmodule",
"order",
"rings_modules_and_algebras",
"divalg",
"poly",
"matrix",
"mxalgebra",
"mxpoly",
"mxred",
"orderedzmod",
"numdomain",
"numfield... | [] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
normalmx_key : pred_key normalmx. | Proof. by []. Qed. | Fact | normalmx_key | algebra | algebra/spectral.v | [
"HB",
"structures",
"mathcomp",
"ssreflect",
"ssrfun",
"ssrbool",
"eqtype",
"ssrnat",
"seq",
"choice",
"fintype",
"bigop",
"nmodule",
"order",
"rings_modules_and_algebras",
"divalg",
"poly",
"matrix",
"mxalgebra",
"mxpoly",
"mxred",
"orderedzmod",
"numdomain",
"numfield... | [
"normalmx"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
normalmx_keyed | := KeyedQualifier normalmx_key. | Canonical | normalmx_keyed | algebra | algebra/spectral.v | [
"HB",
"structures",
"mathcomp",
"ssreflect",
"ssrfun",
"ssrbool",
"eqtype",
"ssrnat",
"seq",
"choice",
"fintype",
"bigop",
"nmodule",
"order",
"rings_modules_and_algebras",
"divalg",
"poly",
"matrix",
"mxalgebra",
"mxpoly",
"mxred",
"orderedzmod",
"numdomain",
"numfield... | [
"normalmx_key"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
normalmxP {M : 'M[C]_n} :
reflect (M *m M ^t* = M ^t* *m M) (M \is normalmx). | Proof. exact: eqP. Qed. | Lemma | normalmxP | algebra | algebra/spectral.v | [
"HB",
"structures",
"mathcomp",
"ssreflect",
"ssrfun",
"ssrbool",
"eqtype",
"ssrnat",
"seq",
"choice",
"fintype",
"bigop",
"nmodule",
"order",
"rings_modules_and_algebras",
"divalg",
"poly",
"matrix",
"mxalgebra",
"mxpoly",
"mxred",
"orderedzmod",
"numdomain",
"numfield... | [
"normalmx"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
hermitian_normalmx (A : 'M[C]_n) : A \is hermsymmx -> A \is normalmx. | Proof.
move=> Ahermi; apply/normalmxP.
by rewrite (trmx_hermitian (HermitianMx Ahermi)) scale1r map_mxCK.
Qed. | Lemma | hermitian_normalmx | algebra | algebra/spectral.v | [
"HB",
"structures",
"mathcomp",
"ssreflect",
"ssrfun",
"ssrbool",
"eqtype",
"ssrnat",
"seq",
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"rings_modules_and_algebras",
"divalg",
"poly",
"matrix",
"mxalgebra",
"mxpoly",
"mxred",
"orderedzmod",
"numdomain",
"numfield... | [
"apply",
"hermsymmx",
"map_mxCK",
"normalmx",
"normalmxP",
"scale1r",
"trmx_hermitian"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
symmetric_normalmx (A : 'M[C]_n) : A \is symmetricmx ->
A \is a realmx -> A \is normalmx. | Proof. by move=> Asym Areal; rewrite hermitian_normalmx// realsym_hermsym. Qed. | Lemma | symmetric_normalmx | algebra | algebra/spectral.v | [
"HB",
"structures",
"mathcomp",
"ssreflect",
"ssrfun",
"ssrbool",
"eqtype",
"ssrnat",
"seq",
"choice",
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"divalg",
"poly",
"matrix",
"mxalgebra",
"mxpoly",
"mxred",
"orderedzmod",
"numdomain",
"numfield... | [
"hermitian_normalmx",
"normalmx",
"realmx",
"realsym_hermsym",
"symmetricmx"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
dotmx_def | := (form_of_matrix (@conjC _) 1%:M). | Notation | dotmx_def | algebra | algebra/spectral.v | [
"HB",
"structures",
"mathcomp",
"ssreflect",
"ssrfun",
"ssrbool",
"eqtype",
"ssrnat",
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"divalg",
"poly",
"matrix",
"mxalgebra",
"mxpoly",
"mxred",
"orderedzmod",
"numdomain",
"numfield... | [
"conjC"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
dotmx (C : numClosedFieldType) n (u v : 'rV[C]_n) | :=
dotmx_def u%R v%R. | Definition | dotmx | algebra | algebra/spectral.v | [
"HB",
"structures",
"mathcomp",
"ssreflect",
"ssrfun",
"ssrbool",
"eqtype",
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"divalg",
"poly",
"matrix",
"mxalgebra",
"mxpoly",
"mxred",
"orderedzmod",
"numdomain",
"numfield... | [
"dotmx_def"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
"''[' u , v ]" | := (dotmx u v) : ring_scope. | Notation | ''[' u , v ] | algebra | algebra/spectral.v | [
"HB",
"structures",
"mathcomp",
"ssreflect",
"ssrfun",
"ssrbool",
"eqtype",
"ssrnat",
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"divalg",
"poly",
"matrix",
"mxalgebra",
"mxpoly",
"mxred",
"orderedzmod",
"numdomain",
"numfield... | [
"dotmx"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
dotmxE n (u v : 'rV[C]_n) : '[u, v] = ( u *m v ^t* ) 0 0. | Proof. by rewrite /dotmx unlock mulmx1 /= trace_mx11. Qed. | Lemma | dotmxE | algebra | algebra/spectral.v | [
"HB",
"structures",
"mathcomp",
"ssreflect",
"ssrfun",
"ssrbool",
"eqtype",
"ssrnat",
"seq",
"choice",
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"order",
"rings_modules_and_algebras",
"divalg",
"poly",
"matrix",
"mxalgebra",
"mxpoly",
"mxred",
"orderedzmod",
"numdomain",
"numfield... | [
"dotmx",
"mulmx1",
"trace_mx11"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
row_unitarymxP m n {M : 'M[C]_(m, n)} :
reflect (forall i j, '[row i M, row j M] = (i == j)%:R) (M \is unitarymx). | Proof.
apply: (iffP eqP) => [Mo i j|Mo].
have /matrixP/(_ i j) := Mo; rewrite !mxE => <-.
by rewrite dotmxE !mxE; apply: eq_bigr => /= k _; rewrite !mxE.
apply/matrixP=> i j; rewrite !mxE; have := Mo i j; rewrite dotmxE !mxE => <-.
by apply: eq_bigr => /= k _; rewrite !mxE.
Qed. | Lemma | row_unitarymxP | algebra | algebra/spectral.v | [
"HB",
"structures",
"mathcomp",
"ssreflect",
"ssrfun",
"ssrbool",
"eqtype",
"ssrnat",
"seq",
"choice",
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"bigop",
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"rings_modules_and_algebras",
"divalg",
"poly",
"matrix",
"mxalgebra",
"mxpoly",
"mxred",
"orderedzmod",
"numdomain",
"numfield... | [
"apply",
"dotmxE",
"eq_bigr",
"matrixP",
"mxE",
"row",
"unitarymx"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
dotmx_is_hermitian n : isHermitianSesquilinear _ _ false conjC (@dotmx C n). | Proof.
split=> /= u v; rewrite !dotmxE/= expr0 mul1r.
suff -> : u *m v ^t* = ((v *m u ^t* ) ^t* ) by rewrite !mxE.
by rewrite !trmx_mul map_mxM/= trmxCK.
Qed. | Fact | dotmx_is_hermitian | algebra | algebra/spectral.v | [
"HB",
"structures",
"mathcomp",
"ssreflect",
"ssrfun",
"ssrbool",
"eqtype",
"ssrnat",
"seq",
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"rings_modules_and_algebras",
"divalg",
"poly",
"matrix",
"mxalgebra",
"mxpoly",
"mxred",
"orderedzmod",
"numdomain",
"numfield... | [
"conjC",
"dotmx",
"dotmxE",
"expr0",
"map_mxM",
"mul1r",
"mxE",
"split",
"trmxCK",
"trmx_mul"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
dotmx_is_dot n : isDotProduct _ _ (@dotmx C n). | Proof.
split => /= u u_neq0; rewrite dotmxE mxE.
suff /existsP[i ui_neq0] : [exists i, u 0%R i != 0].
rewrite (bigD1 i) //= ltr_wpDr// ?sumr_ge0// ?mxE ?mul_conjC_gt0//.
by move=> j _; rewrite !mxE mul_conjC_ge0.
apply: contraNT u_neq0; rewrite negb_exists => /forallP uNN0.
by apply/eqP/rowP=> j; rewrite mxE; apply... | Fact | dotmx_is_dot | algebra | algebra/spectral.v | [
"HB",
"structures",
"mathcomp",
"ssreflect",
"ssrfun",
"ssrbool",
"eqtype",
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"rings_modules_and_algebras",
"divalg",
"poly",
"matrix",
"mxalgebra",
"mxpoly",
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"orderedzmod",
"numdomain",
"numfield... | [
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"existsP",
"forallP",
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"mul_conjC_ge0",
"mul_conjC_gt0",
"mxE",
"negb_exists",
"rowP",
"split",
"sumr_ge0"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
"B ^!" | :=
(orthomx conjC (mx_of_hermitian (hermitian1mx _)) B) :
matrix_set_scope. | Notation | B ^! | algebra | algebra/spectral.v | [
"HB",
"structures",
"mathcomp",
"ssreflect",
"ssrfun",
"ssrbool",
"eqtype",
"ssrnat",
"seq",
"choice",
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"rings_modules_and_algebras",
"divalg",
"poly",
"matrix",
"mxalgebra",
"mxpoly",
"mxred",
"orderedzmod",
"numdomain",
"numfield... | [
"conjC",
"hermitian1mx",
"orthomx"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
"A '_|_ B" | := (A%MS <= B^!)%MS : bool_scope. | Notation | A '_|_ B | algebra | algebra/spectral.v | [
"HB",
"structures",
"mathcomp",
"ssreflect",
"ssrfun",
"ssrbool",
"eqtype",
"ssrnat",
"seq",
"choice",
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"order",
"rings_modules_and_algebras",
"divalg",
"poly",
"matrix",
"mxalgebra",
"mxpoly",
"mxred",
"orderedzmod",
"numdomain",
"numfield... | [] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
orthomx1E m n p (A : 'M[C]_(m, n)) (B : 'M_(p, n)) :
(A '_|_ B)%MS = (A *m B^t* == 0). | Proof. by apply/sub_kermxP/eqP; rewrite !mul1mx. Qed. | Lemma | orthomx1E | algebra | algebra/spectral.v | [
"HB",
"structures",
"mathcomp",
"ssreflect",
"ssrfun",
"ssrbool",
"eqtype",
"ssrnat",
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"divalg",
"poly",
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"mxalgebra",
"mxpoly",
"mxred",
"orderedzmod",
"numdomain",
"numfield... | [
"apply",
"mul1mx",
"sub_kermxP"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
orthomx1P m n p {A : 'M[C]_(m, n)} {B : 'M_(p, n)} :
reflect (A *m B^t* = 0) (A '_|_ B). | Proof. by rewrite orthomx1E; exact/eqP. Qed. | Lemma | orthomx1P | algebra | algebra/spectral.v | [
"HB",
"structures",
"mathcomp",
"ssreflect",
"ssrfun",
"ssrbool",
"eqtype",
"ssrnat",
"seq",
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"order",
"rings_modules_and_algebras",
"divalg",
"poly",
"matrix",
"mxalgebra",
"mxpoly",
"mxred",
"orderedzmod",
"numdomain",
"numfield... | [
"orthomx1E"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
orthomx_disj n p q (A : 'M[C]_(p, n)) (B :'M_(q, n)) :
A '_|_ B -> (A :&: B = 0)%MS. | Proof.
move=> nAB; apply/eqP/rowV0Pn => [[v]]; rewrite sub_capmx => /andP [vA vB].
apply/negP; rewrite negbK -(dnorm_eq0 (@dotmx C n)).
by rewrite -orthomxE (orthomxP _ _ _ nAB).
Qed. | Lemma | orthomx_disj | algebra | algebra/spectral.v | [
"HB",
"structures",
"mathcomp",
"ssreflect",
"ssrfun",
"ssrbool",
"eqtype",
"ssrnat",
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"rings_modules_and_algebras",
"divalg",
"poly",
"matrix",
"mxalgebra",
"mxpoly",
"mxred",
"orderedzmod",
"numdomain",
"numfield... | [
"apply",
"dnorm_eq0",
"dotmx",
"orthomxE",
"orthomxP",
"rowV0Pn",
"sub_capmx",
"vA",
"vB"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
orthomx_ortho_disj n p (A : 'M[C]_(p, n)) : (A :&: A^! = 0)%MS. | Proof. exact/orthomx_disj/ortho_mx_ortho. Qed. | Lemma | orthomx_ortho_disj | algebra | algebra/spectral.v | [
"HB",
"structures",
"mathcomp",
"ssreflect",
"ssrfun",
"ssrbool",
"eqtype",
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"divalg",
"poly",
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"mxalgebra",
"mxpoly",
"mxred",
"orderedzmod",
"numdomain",
"numfield... | [
"ortho_mx_ortho",
"orthomx_disj"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
rank_ortho p n (A : 'M[C]_(p, n)) : \rank A^! = (n - \rank A)%N. | Proof. by rewrite mxrank_ker mul1mx mxrank_map mxrank_tr. Qed. | Lemma | rank_ortho | algebra | algebra/spectral.v | [
"HB",
"structures",
"mathcomp",
"ssreflect",
"ssrfun",
"ssrbool",
"eqtype",
"ssrnat",
"seq",
"choice",
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"rings_modules_and_algebras",
"divalg",
"poly",
"matrix",
"mxalgebra",
"mxpoly",
"mxred",
"orderedzmod",
"numdomain",
"numfield... | [
"mul1mx",
"mxrank_ker",
"mxrank_map",
"mxrank_tr",
"rank"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
add_rank_ortho p n (A : 'M[C]_(p, n)) : (\rank A + \rank A^!)%N = n. | Proof. by rewrite rank_ortho subnKC ?rank_leq_col. Qed. | Lemma | add_rank_ortho | algebra | algebra/spectral.v | [
"HB",
"structures",
"mathcomp",
"ssreflect",
"ssrfun",
"ssrbool",
"eqtype",
"ssrnat",
"seq",
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"rings_modules_and_algebras",
"divalg",
"poly",
"matrix",
"mxalgebra",
"mxpoly",
"mxred",
"orderedzmod",
"numdomain",
"numfield... | [
"rank",
"rank_leq_col",
"rank_ortho",
"subnKC"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
addsmx_ortho p n (A : 'M[C]_(p, n)) : (A + A^! :=: 1%:M)%MS. | Proof.
apply/eqmxP/andP; rewrite submx1; split=> //.
rewrite -mxrank_leqif_sup ?submx1 ?mxrank1 ?(mxdirectP _) /= ?add_rank_ortho //.
by rewrite mxdirect_addsE /= ?mxdirectE ?orthomx_ortho_disj !eqxx.
Qed. | Lemma | addsmx_ortho | algebra | algebra/spectral.v | [
"HB",
"structures",
"mathcomp",
"ssreflect",
"ssrfun",
"ssrbool",
"eqtype",
"ssrnat",
"seq",
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"rings_modules_and_algebras",
"divalg",
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"mxalgebra",
"mxpoly",
"mxred",
"orderedzmod",
"numdomain",
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"apply",
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"mxdirectP",
"mxdirect_addsE",
"mxrank1",
"mxrank_leqif_sup",
"orthomx_ortho_disj",
"split",
"submx1"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
ortho_id p n (A : 'M[C]_(p, n)) : (A^!^! :=: A)%MS. | Proof.
apply/eqmx_sym/eqmxP.
by rewrite -mxrank_leqif_eq 1?orthomx_sym // !rank_ortho subKn // ?rank_leq_col.
Qed. | Lemma | ortho_id | algebra | algebra/spectral.v | [
"HB",
"structures",
"mathcomp",
"ssreflect",
"ssrfun",
"ssrbool",
"eqtype",
"ssrnat",
"seq",
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"rings_modules_and_algebras",
"divalg",
"poly",
"matrix",
"mxalgebra",
"mxpoly",
"mxred",
"orderedzmod",
"numdomain",
"numfield... | [
"apply",
"eqmxP",
"eqmx_sym",
"mxrank_leqif_eq",
"orthomx_sym",
"rank_leq_col",
"rank_ortho",
"subKn"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
submx_ortho p m n (U : 'M[C]_(p, n)) (V : 'M_(m, n)) :
(U^! <= V^!)%MS = (V <= U)%MS. | Proof. by rewrite orthomx_sym ortho_id. Qed. | Lemma | submx_ortho | algebra | algebra/spectral.v | [
"HB",
"structures",
"mathcomp",
"ssreflect",
"ssrfun",
"ssrbool",
"eqtype",
"ssrnat",
"seq",
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"rings_modules_and_algebras",
"divalg",
"poly",
"matrix",
"mxalgebra",
"mxpoly",
"mxred",
"orderedzmod",
"numdomain",
"numfield... | [
"ortho_id",
"orthomx_sym"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
proj_ortho p n (U : 'M[C]_(p, n)) | := proj_mx <<U>>%MS U^!%MS. | Definition | proj_ortho | algebra | algebra/spectral.v | [
"HB",
"structures",
"mathcomp",
"ssreflect",
"ssrfun",
"ssrbool",
"eqtype",
"ssrnat",
"seq",
"choice",
"fintype",
"bigop",
"nmodule",
"order",
"rings_modules_and_algebras",
"divalg",
"poly",
"matrix",
"mxalgebra",
"mxpoly",
"mxred",
"orderedzmod",
"numdomain",
"numfield... | [
"proj_mx"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
sub_adds_genmx_ortho (p m n : nat) (U : 'M[C]_(p, n)) (W : 'M_(m, n)) :
(W <= <<U>> + U^!)%MS. | Proof.
by rewrite !(adds_eqmx (genmxE _) (eqmx_refl _)) addsmx_ortho submx1.
Qed. | Lemma | sub_adds_genmx_ortho | algebra | algebra/spectral.v | [
"HB",
"structures",
"mathcomp",
"ssreflect",
"ssrfun",
"ssrbool",
"eqtype",
"ssrnat",
"seq",
"choice",
"fintype",
"bigop",
"nmodule",
"order",
"rings_modules_and_algebras",
"divalg",
"poly",
"matrix",
"mxalgebra",
"mxpoly",
"mxred",
"orderedzmod",
"numdomain",
"numfield... | [
"adds_eqmx",
"addsmx_ortho",
"eqmx_refl",
"genmxE",
"nat",
"submx1"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
cap_genmx_ortho p n (U : 'M[C]_(p, n)) : (<<U>> :&: U^!)%MS = 0. | Proof.
apply/eqmx0P; rewrite !(cap_eqmx (genmxE _) (eqmx_refl _)).
by rewrite orthomx_ortho_disj; exact/eqmx0P.
Qed. | Lemma | cap_genmx_ortho | algebra | algebra/spectral.v | [
"HB",
"structures",
"mathcomp",
"ssreflect",
"ssrfun",
"ssrbool",
"eqtype",
"ssrnat",
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"rings_modules_and_algebras",
"divalg",
"poly",
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"mxalgebra",
"mxpoly",
"mxred",
"orderedzmod",
"numdomain",
"numfield... | [
"apply",
"cap_eqmx",
"eqmx0P",
"eqmx_refl",
"genmxE",
"orthomx_ortho_disj"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
proj_ortho_sub p m n (U : 'M_(p, n)) (W : 'M_(m, n)) :
(W *m proj_ortho U <= U)%MS. | Proof. by rewrite (submx_trans (proj_mx_sub _ _ _)) // genmxE. Qed. | Lemma | proj_ortho_sub | algebra | algebra/spectral.v | [
"HB",
"structures",
"mathcomp",
"ssreflect",
"ssrfun",
"ssrbool",
"eqtype",
"ssrnat",
"seq",
"choice",
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"order",
"rings_modules_and_algebras",
"divalg",
"poly",
"matrix",
"mxalgebra",
"mxpoly",
"mxred",
"orderedzmod",
"numdomain",
"numfield... | [
"genmxE",
"proj_mx_sub",
"proj_ortho",
"submx_trans"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
proj_ortho_compl_sub p m n (U : 'M_(p, n)) (W : 'M_(m, n)) :
(W - W *m proj_ortho U <= U^!)%MS. | Proof. by rewrite proj_mx_compl_sub // addsmx_ortho submx1. Qed. | Lemma | proj_ortho_compl_sub | algebra | algebra/spectral.v | [
"HB",
"structures",
"mathcomp",
"ssreflect",
"ssrfun",
"ssrbool",
"eqtype",
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"numdomain",
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"proj_mx_compl_sub",
"proj_ortho",
"submx1"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
proj_ortho_id p m n (U : 'M_(p, n)) (W : 'M_(m, n)) :
(W <= U)%MS -> W *m proj_ortho U = W. | Proof. by move=> WU; rewrite proj_mx_id ?genmxE. Qed. | Lemma | proj_ortho_id | algebra | algebra/spectral.v | [
"HB",
"structures",
"mathcomp",
"ssreflect",
"ssrfun",
"ssrbool",
"eqtype",
"ssrnat",
"seq",
"choice",
"fintype",
"bigop",
"nmodule",
"order",
"rings_modules_and_algebras",
"divalg",
"poly",
"matrix",
"mxalgebra",
"mxpoly",
"mxred",
"orderedzmod",
"numdomain",
"numfield... | [
"genmxE",
"proj_mx_id",
"proj_ortho"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
proj_ortho_0 p m n (U : 'M_(p, n)) (W : 'M_(m, n)) :
(W <= U^!)%MS -> W *m proj_ortho U = 0. | Proof. by move=> WUo; rewrite proj_mx_0. Qed. | Lemma | proj_ortho_0 | algebra | algebra/spectral.v | [
"HB",
"structures",
"mathcomp",
"ssreflect",
"ssrfun",
"ssrbool",
"eqtype",
"ssrnat",
"seq",
"choice",
"fintype",
"bigop",
"nmodule",
"order",
"rings_modules_and_algebras",
"divalg",
"poly",
"matrix",
"mxalgebra",
"mxpoly",
"mxred",
"orderedzmod",
"numdomain",
"numfield... | [
"proj_mx_0",
"proj_ortho"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
add_proj_ortho p m n (U : 'M_(p, n)) (W : 'M_(m, n)) :
W *m proj_ortho U + W *m proj_ortho U^!%MS = W. | Proof.
rewrite -[W in LHS](@add_proj_mx _ _ _ <<U>>%MS U^!%MS W)//.
rewrite !mulmxDl proj_ortho_id ?proj_ortho_sub //.
rewrite proj_ortho_0 ?proj_mx_sub // addr0.
rewrite proj_ortho_0 ?ortho_id ?proj_ortho_sub // add0r.
by rewrite proj_ortho_id ?proj_mx_sub// add_proj_mx.
Qed. | Lemma | add_proj_ortho | algebra | algebra/spectral.v | [
"HB",
"structures",
"mathcomp",
"ssreflect",
"ssrfun",
"ssrbool",
"eqtype",
"ssrnat",
"seq",
"choice",
"fintype",
"bigop",
"nmodule",
"order",
"rings_modules_and_algebras",
"divalg",
"poly",
"matrix",
"mxalgebra",
"mxpoly",
"mxred",
"orderedzmod",
"numdomain",
"numfield... | [
"add0r",
"add_proj_mx",
"addr0",
"mulmxDl",
"ortho_id",
"proj_mx_sub",
"proj_ortho",
"proj_ortho_0",
"proj_ortho_id",
"proj_ortho_sub"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
proj_ortho_proj m n (U : 'M_(m, n)) : let P := proj_ortho U in P *m P = P. | Proof. by rewrite /= proj_mx_proj. Qed. | Lemma | proj_ortho_proj | algebra | algebra/spectral.v | [
"HB",
"structures",
"mathcomp",
"ssreflect",
"ssrfun",
"ssrbool",
"eqtype",
"ssrnat",
"seq",
"choice",
"fintype",
"bigop",
"nmodule",
"order",
"rings_modules_and_algebras",
"divalg",
"poly",
"matrix",
"mxalgebra",
"mxpoly",
"mxred",
"orderedzmod",
"numdomain",
"numfield... | [
"proj_mx_proj",
"proj_ortho"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
proj_orthoE p n (U : 'M_(p, n)) : (proj_ortho U :=: U)%MS. | Proof.
apply/eqmxP/andP; split; first by rewrite -proj_ortho_proj proj_ortho_sub.
by rewrite -[X in (X <= _)%MS](proj_ortho_id (submx_refl U)) mulmx_sub.
Qed. | Lemma | proj_orthoE | algebra | algebra/spectral.v | [
"HB",
"structures",
"mathcomp",
"ssreflect",
"ssrfun",
"ssrbool",
"eqtype",
"ssrnat",
"seq",
"choice",
"fintype",
"bigop",
"nmodule",
"order",
"rings_modules_and_algebras",
"divalg",
"poly",
"matrix",
"mxalgebra",
"mxpoly",
"mxred",
"orderedzmod",
"numdomain",
"numfield... | [
"apply",
"eqmxP",
"mulmx_sub",
"proj_ortho",
"proj_ortho_id",
"proj_ortho_proj",
"proj_ortho_sub",
"split",
"submx_refl"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
orthomx_proj_mx_ortho p p' m m' n
(A : 'M_(p, n)) (A' : 'M_(p', n))
(W : 'M_(m, n)) (W' : 'M_(m', n)) :
A '_|_ A' -> W *m proj_ortho A '_|_ W' *m proj_ortho A'. | Proof.
rewrite orthomx_sym => An.
rewrite mulmx_sub // orthomx_sym (eqmx_ortho _ (proj_orthoE _)).
by rewrite (submx_trans _ An) // proj_ortho_sub.
Qed. | Lemma | orthomx_proj_mx_ortho | algebra | algebra/spectral.v | [
"HB",
"structures",
"mathcomp",
"ssreflect",
"ssrfun",
"ssrbool",
"eqtype",
"ssrnat",
"seq",
"choice",
"fintype",
"bigop",
"nmodule",
"order",
"rings_modules_and_algebras",
"divalg",
"poly",
"matrix",
"mxalgebra",
"mxpoly",
"mxred",
"orderedzmod",
"numdomain",
"numfield... | [
"A'",
"eqmx_ortho",
"mulmx_sub",
"orthomx_sym",
"proj_ortho",
"proj_orthoE",
"proj_ortho_sub",
"submx_trans"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
schmidt_subproof m n (A : 'M[C]_(m, n)) : (m <= n)%N ->
exists2 B : 'M_(m, n), B \is unitarymx & [forall i : 'I_m,
(row i A <= (\sum_(k < m | (k <= i)%N) <<row k B>>))%MS
&& ('[row i A, row i B] >= 0) ]. | Proof.
elim: m A => [|m IHm].
exists (pid_mx n); first by rewrite qualifE !thinmx0.
by apply/forallP=> -[].
rewrite -addn1 => A leq_Sm_n.
have lemSm : (m <= m + 1)%N by rewrite addn1.
have ltmSm : (m < m + 1)%N by rewrite addn1.
have lemn : (m <= n)%N by rewrite ltnW // -addn1.
have [B Bortho] := IHm (usubmx A) lem... | Lemma | schmidt_subproof | algebra | algebra/spectral.v | [
"HB",
"structures",
"mathcomp",
"ssreflect",
"ssrfun",
"ssrbool",
"eqtype",
"ssrnat",
"seq",
"choice",
"fintype",
"bigop",
"nmodule",
"order",
"rings_modules_and_algebras",
"divalg",
"poly",
"matrix",
"mxalgebra",
"mxpoly",
"mxred",
"orderedzmod",
"numdomain",
"numfield... | [
"add0r",
"add_proj_ortho",
"addmx_sub_adds",
"addn0",
"addn1",
"addr0",
"addsmxC",
"addsmxS",
"apply",
"bigD1",
"col_mx",
"conjC_ge0",
"dnormZ",
"dnorm_eq0",
"dnorm_ge0",
"dnorm_gt0",
"dsubmx",
"eqVneq",
"eq_bigr",
"eq_shift",
"eqmx_scale",
"exprVn",
"forallP",
"genmxE"... | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
schmidt m n (A : 'M[C]_(m, n)) | :=
if (m <= n)%N =P true is ReflectT le_mn
then projT1 (sig2_eqW (schmidt_subproof A le_mn))
else A. | Definition | schmidt | algebra | algebra/spectral.v | [
"HB",
"structures",
"mathcomp",
"ssreflect",
"ssrfun",
"ssrbool",
"eqtype",
"ssrnat",
"seq",
"choice",
"fintype",
"bigop",
"nmodule",
"order",
"rings_modules_and_algebras",
"divalg",
"poly",
"matrix",
"mxalgebra",
"mxpoly",
"mxred",
"orderedzmod",
"numdomain",
"numfield... | [
"schmidt_subproof",
"sig2_eqW"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
schmidt_unitarymx m n (A : 'M[C]_(m, n)) : (m <= n)%N ->
schmidt A \is unitarymx. | Proof. by rewrite /schmidt; case: eqP => // ?; case: sig2_eqW. Qed. | Lemma | schmidt_unitarymx | algebra | algebra/spectral.v | [
"HB",
"structures",
"mathcomp",
"ssreflect",
"ssrfun",
"ssrbool",
"eqtype",
"ssrnat",
"seq",
"choice",
"fintype",
"bigop",
"nmodule",
"order",
"rings_modules_and_algebras",
"divalg",
"poly",
"matrix",
"mxalgebra",
"mxpoly",
"mxred",
"orderedzmod",
"numdomain",
"numfield... | [
"schmidt",
"sig2_eqW",
"unitarymx"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
row_schmidt_sub m n (A : 'M[C]_(m, n)) i :
(row i A <= (\sum_(k < m | (k <= i)%N) <<row k (schmidt A)>>))%MS. | Proof.
rewrite /schmidt; case: eqP => // ?.
by case: sig2_eqW => ? ? /= /forallP /(_ i) /andP[].
by apply/(sumsmx_sup i) => //; rewrite genmxE.
Qed. | Lemma | row_schmidt_sub | algebra | algebra/spectral.v | [
"HB",
"structures",
"mathcomp",
"ssreflect",
"ssrfun",
"ssrbool",
"eqtype",
"ssrnat",
"seq",
"choice",
"fintype",
"bigop",
"nmodule",
"order",
"rings_modules_and_algebras",
"divalg",
"poly",
"matrix",
"mxalgebra",
"mxpoly",
"mxred",
"orderedzmod",
"numdomain",
"numfield... | [
"apply",
"forallP",
"genmxE",
"row",
"schmidt",
"sig2_eqW",
"sumsmx_sup"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
form1_row_schmidt m n (A : 'M[C]_(m, n)) i :
'[row i A, row i (schmidt A)] >= 0. | Proof.
rewrite /schmidt; case: eqP => // ?; rewrite ?dnorm_ge0 //.
by case: sig2_eqW => ? ? /= /forallP /(_ i) /andP[].
Qed. | Lemma | form1_row_schmidt | algebra | algebra/spectral.v | [
"HB",
"structures",
"mathcomp",
"ssreflect",
"ssrfun",
"ssrbool",
"eqtype",
"ssrnat",
"seq",
"choice",
"fintype",
"bigop",
"nmodule",
"order",
"rings_modules_and_algebras",
"divalg",
"poly",
"matrix",
"mxalgebra",
"mxpoly",
"mxred",
"orderedzmod",
"numdomain",
"numfield... | [
"dnorm_ge0",
"forallP",
"row",
"schmidt",
"sig2_eqW"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
schmidt_sub m n (A : 'M[C]_(m, n)) : (A <= schmidt A)%MS. | Proof.
apply/row_subP => i; rewrite (submx_trans (row_schmidt_sub _ _)) //.
by apply/sumsmx_subP => /= j le_ji; rewrite genmxE row_sub.
Qed. | Lemma | schmidt_sub | algebra | algebra/spectral.v | [
"HB",
"structures",
"mathcomp",
"ssreflect",
"ssrfun",
"ssrbool",
"eqtype",
"ssrnat",
"seq",
"choice",
"fintype",
"bigop",
"nmodule",
"order",
"rings_modules_and_algebras",
"divalg",
"poly",
"matrix",
"mxalgebra",
"mxpoly",
"mxred",
"orderedzmod",
"numdomain",
"numfield... | [
"apply",
"genmxE",
"row_schmidt_sub",
"row_sub",
"row_subP",
"schmidt",
"submx_trans",
"sumsmx_subP"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
eqmx_schmidt_full m n (A : 'M[C]_(m, n)) :
row_full A -> (schmidt A :=: A)%MS. | Proof.
move=> Afull; apply/eqmx_sym/eqmxP; rewrite -mxrank_leqif_eq //.
by rewrite eqn_leq mxrankS //= (@leq_trans n) ?rank_leq_col ?col_leq_rank.
Qed. | Lemma | eqmx_schmidt_full | algebra | algebra/spectral.v | [
"HB",
"structures",
"mathcomp",
"ssreflect",
"ssrfun",
"ssrbool",
"eqtype",
"ssrnat",
"seq",
"choice",
"fintype",
"bigop",
"nmodule",
"order",
"rings_modules_and_algebras",
"divalg",
"poly",
"matrix",
"mxalgebra",
"mxpoly",
"mxred",
"orderedzmod",
"numdomain",
"numfield... | [
"apply",
"col_leq_rank",
"eqmxP",
"eqmx_sym",
"eqn_leq",
"leq_trans",
"mxrankS",
"mxrank_leqif_eq",
"rank_leq_col",
"row_full",
"schmidt"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
eqmx_schmidt_free m n (A : 'M[C]_(m, n)) :
row_free A -> (schmidt A :=: A)%MS. | Proof.
move=> Afree; apply/eqmx_sym/eqmxP; rewrite -mxrank_leqif_eq //.
by rewrite eqn_leq mxrankS //= (@leq_trans m) ?rank_leq_row // ?row_leq_rank.
Qed. | Lemma | eqmx_schmidt_free | algebra | algebra/spectral.v | [
"HB",
"structures",
"mathcomp",
"ssreflect",
"ssrfun",
"ssrbool",
"eqtype",
"ssrnat",
"seq",
"choice",
"fintype",
"bigop",
"nmodule",
"order",
"rings_modules_and_algebras",
"divalg",
"poly",
"matrix",
"mxalgebra",
"mxpoly",
"mxred",
"orderedzmod",
"numdomain",
"numfield... | [
"apply",
"eqmxP",
"eqmx_sym",
"eqn_leq",
"leq_trans",
"mxrankS",
"mxrank_leqif_eq",
"rank_leq_row",
"row_free",
"row_leq_rank",
"schmidt"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
schmidt_complete m n (V : 'M[C]_(m, n)) | :=
col_mx (schmidt (row_base V)) (schmidt (row_base V^!%MS)). | Definition | schmidt_complete | algebra | algebra/spectral.v | [
"HB",
"structures",
"mathcomp",
"ssreflect",
"ssrfun",
"ssrbool",
"eqtype",
"ssrnat",
"seq",
"choice",
"fintype",
"bigop",
"nmodule",
"order",
"rings_modules_and_algebras",
"divalg",
"poly",
"matrix",
"mxalgebra",
"mxpoly",
"mxred",
"orderedzmod",
"numdomain",
"numfield... | [
"col_mx",
"row_base",
"schmidt"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
schmidt_complete_unitarymx m n (V : 'M[C]_(m, n)) :
schmidt_complete V \is unitarymx. | Proof.
apply/unitarymxP; rewrite tr_col_mx map_row_mx mul_col_row.
rewrite !(unitarymxP _) ?schmidt_unitarymx ?rank_leq_col //.
move=> [:nsV]; rewrite !(orthomx1P _) -?scalar_mx_block //;
[abstract: nsV|]; last by rewrite orthomx_sym.
by do 2!rewrite eqmx_schmidt_free ?eq_row_base ?row_base_free // orthomx_sym.
Qed. | Lemma | schmidt_complete_unitarymx | algebra | algebra/spectral.v | [
"HB",
"structures",
"mathcomp",
"ssreflect",
"ssrfun",
"ssrbool",
"eqtype",
"ssrnat",
"seq",
"choice",
"fintype",
"bigop",
"nmodule",
"order",
"rings_modules_and_algebras",
"divalg",
"poly",
"matrix",
"mxalgebra",
"mxpoly",
"mxred",
"orderedzmod",
"numdomain",
"numfield... | [
"apply",
"eq_row_base",
"eqmx_schmidt_free",
"last",
"map_row_mx",
"mul_col_row",
"orthomx1P",
"orthomx_sym",
"rank_leq_col",
"row_base_free",
"scalar_mx_block",
"schmidt_complete",
"schmidt_unitarymx",
"tr_col_mx",
"unitarymx",
"unitarymxP"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
cotrigonalization n (As : seq 'M[C]_n) :
{in As &, forall A B, comm_mx A B} ->
cotrigonalizable_in (@unitarymx C n n) As. | Proof.
elim: n {-2}n (leqnn n) As => [|N IHN] n.
rewrite leqn0 => /eqP n_eq0.
exists 1%:M; first by rewrite qualifE mul1mx trmx1 map_mx1.
apply/allP => ? ?; apply/is_trig_mxP => i j.
by suff: False by []; move: i; rewrite n_eq0 => -[].
rewrite leq_eqVlt => /predU1P [n_eqSN|/IHN//].
have /andP [n_gt0 n_small] : ... | Lemma | cotrigonalization | algebra | algebra/spectral.v | [
"HB",
"structures",
"mathcomp",
"ssreflect",
"ssrfun",
"ssrbool",
"eqtype",
"ssrnat",
"seq",
"choice",
"fintype",
"bigop",
"nmodule",
"order",
"rings_modules_and_algebras",
"divalg",
"poly",
"matrix",
"mxalgebra",
"mxpoly",
"mxred",
"orderedzmod",
"numdomain",
"numfield... | [
"A'",
"False",
"add0r",
"add_rank_ortho",
"addnS",
"addr0",
"all",
"allP",
"apply",
"block_mx",
"block_mxKdr",
"block_mxKur",
"comm_mx",
"common_eigenvector",
"conjC",
"conjC1",
"conjumx",
"cotrigonalizable_in",
"drsubmx",
"eigenspace",
"eigenspaceP",
"eigenvectorP",
"eq_... | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
Schur n (A : 'M[C]_n) : (n > 0)%N ->
trigonalizable_in (@unitarymx C n n) A. | Proof.
case: n => [//|n] in A * => _; have [] := @cotrigonalization _ [:: A].
by move=> ? ? /=; rewrite !in_cons !orbF => /eqP-> /eqP->.
by move=> P P_unitary /=; rewrite andbT=> A_trigo; exists P.
Qed. | Theorem | Schur | algebra | algebra/spectral.v | [
"HB",
"structures",
"mathcomp",
"ssreflect",
"ssrfun",
"ssrbool",
"eqtype",
"ssrnat",
"seq",
"choice",
"fintype",
"bigop",
"nmodule",
"order",
"rings_modules_and_algebras",
"divalg",
"poly",
"matrix",
"mxalgebra",
"mxpoly",
"mxred",
"orderedzmod",
"numdomain",
"numfield... | [
"cotrigonalization",
"in_cons",
"trigonalizable_in",
"unitarymx"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
cotrigonalization2 n (A B : 'M[C]_n) : A *m B = B *m A ->
exists2 P : 'M[C]_n, P \is unitarymx &
similar_trig P A && similar_trig P B. | Proof.
move=> AB_comm; have [] := @cotrigonalization _ [:: A; B].
by move=> ??; rewrite !inE => /orP[]/eqP->/orP[]/eqP->.
move=> P Punitary /allP /= PP; exists P => //.
by rewrite !PP ?(mem_head, in_cons, orbT).
Qed. | Lemma | cotrigonalization2 | algebra | algebra/spectral.v | [
"HB",
"structures",
"mathcomp",
"ssreflect",
"ssrfun",
"ssrbool",
"eqtype",
"ssrnat",
"seq",
"choice",
"fintype",
"bigop",
"nmodule",
"order",
"rings_modules_and_algebras",
"divalg",
"poly",
"matrix",
"mxalgebra",
"mxpoly",
"mxred",
"orderedzmod",
"numdomain",
"numfield... | [
"allP",
"cotrigonalization",
"inE",
"in_cons",
"mem_head",
"similar_trig",
"unitarymx"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
orthomx_spectral_subproof n {A : 'M[C]_n} : reflect
(exists2 sp : 'M_n * 'rV_n,
sp.1 \is unitarymx &
A = invmx sp.1 *m diag_mx sp.2 *m sp.1)
(A \is normalmx). | Proof.
apply: (iffP normalmxP); last first.
move=> [[/= P D] P_unitary ->].
rewrite !trmx_mul !map_mxM !mulmxA invmx_unitary //.
rewrite !trmxCK ![_ *m P *m _]mulmxtVK //.
by rewrite -[X in X *m P]mulmxA tr_diag_mx map_diag_mx diag_mxC mulmxA.
move=> /cotrigonalization2 [P Punitary /andP[]] PA PATC.
have Punit ... | Theorem | orthomx_spectral_subproof | algebra | algebra/spectral.v | [
"HB",
"structures",
"mathcomp",
"ssreflect",
"ssrfun",
"ssrbool",
"eqtype",
"ssrnat",
"seq",
"choice",
"fintype",
"bigop",
"nmodule",
"order",
"rings_modules_and_algebras",
"divalg",
"poly",
"matrix",
"mxalgebra",
"mxpoly",
"mxred",
"orderedzmod",
"numdomain",
"numfield... | [
"apply",
"conjC_eq0",
"conjVmx",
"conjumx",
"cotrigonalization2",
"diag_mx",
"diag_mxC",
"invmx",
"invmx_unitary",
"is_trig_mxP",
"last",
"ltngtP",
"map_diag_mx",
"map_mxM",
"mulmxA",
"mulmxtVK",
"mxE",
"normalmx",
"normalmxP",
"similarLR",
"similar_diag",
"similar_diagPex"... | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
spectralmx n (A : 'M[C]_n) : 'M[C]_n | :=
if @orthomx_spectral_subproof _ A is ReflectT P
then (projT1 (sig2_eqW P)).1 else 1%:M. | Definition | spectralmx | algebra | algebra/spectral.v | [
"HB",
"structures",
"mathcomp",
"ssreflect",
"ssrfun",
"ssrbool",
"eqtype",
"ssrnat",
"seq",
"choice",
"fintype",
"bigop",
"nmodule",
"order",
"rings_modules_and_algebras",
"divalg",
"poly",
"matrix",
"mxalgebra",
"mxpoly",
"mxred",
"orderedzmod",
"numdomain",
"numfield... | [
"orthomx_spectral_subproof",
"sig2_eqW"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
spectral_diag n (A : 'M[C]_n) : 'rV_n | :=
if @orthomx_spectral_subproof _ A is ReflectT P
then (projT1 (sig2_eqW P)).2 else 0. | Definition | spectral_diag | algebra | algebra/spectral.v | [
"HB",
"structures",
"mathcomp",
"ssreflect",
"ssrfun",
"ssrbool",
"eqtype",
"ssrnat",
"seq",
"choice",
"fintype",
"bigop",
"nmodule",
"order",
"rings_modules_and_algebras",
"divalg",
"poly",
"matrix",
"mxalgebra",
"mxpoly",
"mxred",
"orderedzmod",
"numdomain",
"numfield... | [
"orthomx_spectral_subproof",
"sig2_eqW"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
spectral_unitarymx n (A : 'M[C]_n) : spectralmx A \is unitarymx. | Proof.
rewrite /spectralmx; case: orthomx_spectral_subproof; last first.
by move=> _; apply/unitarymxP; rewrite trmx1 map_mx1 mulmx1.
by move=> ?; case: sig2_eqW.
Qed. | Lemma | spectral_unitarymx | algebra | algebra/spectral.v | [
"HB",
"structures",
"mathcomp",
"ssreflect",
"ssrfun",
"ssrbool",
"eqtype",
"ssrnat",
"seq",
"choice",
"fintype",
"bigop",
"nmodule",
"order",
"rings_modules_and_algebras",
"divalg",
"poly",
"matrix",
"mxalgebra",
"mxpoly",
"mxred",
"orderedzmod",
"numdomain",
"numfield... | [
"apply",
"last",
"map_mx1",
"mulmx1",
"orthomx_spectral_subproof",
"sig2_eqW",
"spectralmx",
"trmx1",
"unitarymx",
"unitarymxP"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
spectral_unit n (A : 'M[C]_n) : spectralmx A \in unitmx. | Proof. exact/unitarymx_unit/spectral_unitarymx. Qed. | Lemma | spectral_unit | algebra | algebra/spectral.v | [
"HB",
"structures",
"mathcomp",
"ssreflect",
"ssrfun",
"ssrbool",
"eqtype",
"ssrnat",
"seq",
"choice",
"fintype",
"bigop",
"nmodule",
"order",
"rings_modules_and_algebras",
"divalg",
"poly",
"matrix",
"mxalgebra",
"mxpoly",
"mxred",
"orderedzmod",
"numdomain",
"numfield... | [
"spectral_unitarymx",
"spectralmx",
"unitarymx_unit",
"unitmx"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
orthomx_spectralP {n} {A : 'M[C]_n}
(P := spectralmx A) (sp := spectral_diag A) :
reflect (A = invmx P *m diag_mx sp *m P) (A \is normalmx). | Proof.
rewrite /P /sp /spectralmx /spectral_diag.
case: orthomx_spectral_subproof.
by move=> Psp; case: sig2_eqW => //=; constructor.
move=> /orthomx_spectral_subproof Ann; constructor; apply/eqP.
apply: contra Ann; rewrite invmx1 mul1mx mulmx1 => /eqP->.
suff -> : diag_mx 0 = 0 by rewrite qualifE trmx0 (map_mx0 conj... | Theorem | orthomx_spectralP | algebra | algebra/spectral.v | [
"HB",
"structures",
"mathcomp",
"ssreflect",
"ssrfun",
"ssrbool",
"eqtype",
"ssrnat",
"seq",
"choice",
"fintype",
"bigop",
"nmodule",
"order",
"rings_modules_and_algebras",
"divalg",
"poly",
"matrix",
"mxalgebra",
"mxpoly",
"mxred",
"orderedzmod",
"numdomain",
"numfield... | [
"apply",
"conjC",
"diag_mx",
"invmx",
"invmx1",
"map_mx0",
"matrixP",
"mul0rn",
"mul1mx",
"mulmx1",
"mxE",
"normalmx",
"orthomx_spectral_subproof",
"sig2_eqW",
"sp",
"spectral_diag",
"spectralmx",
"trmx0"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
hermitian_spectral_diag_real n (A : 'M[C]_n) : A \is hermsymmx ->
spectral_diag A \is a realmx. | Proof.
move=> Ahermi; have /hermitian_normalmx /orthomx_spectralP A_eq := Ahermi.
have /(congr1 ( fun X => X^t* )) := A_eq.
rewrite invmx_unitary ?spectral_unitarymx //.
rewrite !trmx_mul !map_mxM map_trmx trmxK -map_mx_comp.
rewrite tr_diag_mx map_diag_mx (map_mx_id (@conjCK _)).
rewrite -[in RHS]invmx_unitary ?spectr... | Lemma | hermitian_spectral_diag_real | algebra | algebra/spectral.v | [
"HB",
"structures",
"mathcomp",
"ssreflect",
"ssrfun",
"ssrbool",
"eqtype",
"ssrnat",
"seq",
"choice",
"fintype",
"bigop",
"nmodule",
"order",
"rings_modules_and_algebras",
"divalg",
"poly",
"matrix",
"mxalgebra",
"mxpoly",
"mxred",
"orderedzmod",
"numdomain",
"numfield... | [
"CrealP",
"apply",
"conjCK",
"eqxx",
"expr0",
"hermitian_normalmx",
"hermsymmx",
"invmx",
"invmx_unitary",
"is_hermitianmxP",
"map_diag_mx",
"map_mxM",
"map_mx_comp",
"map_mx_id",
"map_trmx",
"matrixP",
"mulKVmx",
"mulmx",
"mulmxA",
"mulmxK",
"mulr1n",
"mxE",
"mxOverP",
... | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
int : Set | := Posz of nat | Negz of nat. | Variant | int | algebra | algebra/ssrint.v | [
"HB",
"structures",
"mathcomp",
"ssreflect",
"ssrfun",
"ssrbool",
"eqtype",
"ssrnat",
"choice",
"seq",
"fintype",
"finfun",
"bigop",
"order",
"nmodule",
"rings_modules_and_algebras",
"divalg",
"countalg",
"poly",
"orderedzmod",
"numdomain",
"numfield",
"Order.TTheory",
... | [
"Posz",
"nat"
] | Defining int | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d |
"n %:Z" | := (Posz n) (only parsing) : int_scope. | Notation | n %:Z | algebra | algebra/ssrint.v | [
"HB",
"structures",
"mathcomp",
"ssreflect",
"ssrfun",
"ssrbool",
"eqtype",
"ssrnat",
"choice",
"seq",
"fintype",
"finfun",
"bigop",
"order",
"nmodule",
"rings_modules_and_algebras",
"divalg",
"countalg",
"poly",
"orderedzmod",
"numdomain",
"numfield",
"Order.TTheory",
... | [
"Posz"
] | Coercion Posz : nat >-> int. | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d |
"n %:Z" | := (Posz n) (only parsing) : ring_scope. | Notation | n %:Z | algebra | algebra/ssrint.v | [
"HB",
"structures",
"mathcomp",
"ssreflect",
"ssrfun",
"ssrbool",
"eqtype",
"ssrnat",
"choice",
"seq",
"fintype",
"finfun",
"bigop",
"order",
"nmodule",
"rings_modules_and_algebras",
"divalg",
"countalg",
"poly",
"orderedzmod",
"numdomain",
"numfield",
"Order.TTheory",
... | [
"Posz"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
"n = m :> 'int'" | := (@eq int n%Z m%Z) (only parsing) : ring_scope. | Notation | n = m :> 'int' | algebra | algebra/ssrint.v | [
"HB",
"structures",
"mathcomp",
"ssreflect",
"ssrfun",
"ssrbool",
"eqtype",
"ssrnat",
"choice",
"seq",
"fintype",
"finfun",
"bigop",
"order",
"nmodule",
"rings_modules_and_algebras",
"divalg",
"countalg",
"poly",
"orderedzmod",
"numdomain",
"numfield",
"Order.TTheory",
... | [
"int"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
"n = m :> 'int'" | := (Posz n = Posz m) (only printing) : ring_scope. | Notation | n = m :> 'int' | algebra | algebra/ssrint.v | [
"HB",
"structures",
"mathcomp",
"ssreflect",
"ssrfun",
"ssrbool",
"eqtype",
"ssrnat",
"choice",
"seq",
"fintype",
"finfun",
"bigop",
"order",
"nmodule",
"rings_modules_and_algebras",
"divalg",
"countalg",
"poly",
"orderedzmod",
"numdomain",
"numfield",
"Order.TTheory",
... | [
"Posz"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
"n == m :> 'int'" | := ((n%Z : int) == (m%Z : int)) (only parsing)
: ring_scope. | Notation | n == m :> 'int' | algebra | algebra/ssrint.v | [
"HB",
"structures",
"mathcomp",
"ssreflect",
"ssrfun",
"ssrbool",
"eqtype",
"ssrnat",
"choice",
"seq",
"fintype",
"finfun",
"bigop",
"order",
"nmodule",
"rings_modules_and_algebras",
"divalg",
"countalg",
"poly",
"orderedzmod",
"numdomain",
"numfield",
"Order.TTheory",
... | [
"int"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
"n == m :> 'int'" | := (Posz n == Posz m) (only printing) : ring_scope. | Notation | n == m :> 'int' | algebra | algebra/ssrint.v | [
"HB",
"structures",
"mathcomp",
"ssreflect",
"ssrfun",
"ssrbool",
"eqtype",
"ssrnat",
"choice",
"seq",
"fintype",
"finfun",
"bigop",
"order",
"nmodule",
"rings_modules_and_algebras",
"divalg",
"countalg",
"poly",
"orderedzmod",
"numdomain",
"numfield",
"Order.TTheory",
... | [
"Posz"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
"n != m :> 'int'" | := ((n%Z : int) != (m%Z : int)) (only parsing)
: ring_scope. | Notation | n != m :> 'int' | algebra | algebra/ssrint.v | [
"HB",
"structures",
"mathcomp",
"ssreflect",
"ssrfun",
"ssrbool",
"eqtype",
"ssrnat",
"choice",
"seq",
"fintype",
"finfun",
"bigop",
"order",
"nmodule",
"rings_modules_and_algebras",
"divalg",
"countalg",
"poly",
"orderedzmod",
"numdomain",
"numfield",
"Order.TTheory",
... | [
"int"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
"n != m :> 'int'" | := (Posz n != Posz m) (only printing) : ring_scope. | Notation | n != m :> 'int' | algebra | algebra/ssrint.v | [
"HB",
"structures",
"mathcomp",
"ssreflect",
"ssrfun",
"ssrbool",
"eqtype",
"ssrnat",
"choice",
"seq",
"fintype",
"finfun",
"bigop",
"order",
"nmodule",
"rings_modules_and_algebras",
"divalg",
"countalg",
"poly",
"orderedzmod",
"numdomain",
"numfield",
"Order.TTheory",
... | [
"Posz"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
"n <> m :> 'int'" | := (not (@eq int n%Z m%Z)) (only parsing)
: ring_scope. | Notation | n <> m :> 'int' | algebra | algebra/ssrint.v | [
"HB",
"structures",
"mathcomp",
"ssreflect",
"ssrfun",
"ssrbool",
"eqtype",
"ssrnat",
"choice",
"seq",
"fintype",
"finfun",
"bigop",
"order",
"nmodule",
"rings_modules_and_algebras",
"divalg",
"countalg",
"poly",
"orderedzmod",
"numdomain",
"numfield",
"Order.TTheory",
... | [
"int"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
"n <> m :> 'int'" | := (Posz n <> Posz m) (only printing) : ring_scope. | Notation | n <> m :> 'int' | algebra | algebra/ssrint.v | [
"HB",
"structures",
"mathcomp",
"ssreflect",
"ssrfun",
"ssrbool",
"eqtype",
"ssrnat",
"choice",
"seq",
"fintype",
"finfun",
"bigop",
"order",
"nmodule",
"rings_modules_and_algebras",
"divalg",
"countalg",
"poly",
"orderedzmod",
"numdomain",
"numfield",
"Order.TTheory",
... | [
"Posz"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
parse_int (x : Number.int) : int | :=
match x with
| Number.IntDecimal (Decimal.Pos u) => Posz (Nat.of_uint u)
| Number.IntDecimal (Decimal.Neg u) => Negz (Nat.of_uint u).-1
| Number.IntHexadecimal (Hexadecimal.Pos u) => Posz (Nat.of_hex_uint u)
| Number.IntHexadecimal (Hexadecimal.Neg u) => Negz (Nat.of_hex_uint u).-1
end. | Definition | parse_int | algebra | algebra/ssrint.v | [
"HB",
"structures",
"mathcomp",
"ssreflect",
"ssrfun",
"ssrbool",
"eqtype",
"ssrnat",
"choice",
"seq",
"fintype",
"finfun",
"bigop",
"order",
"nmodule",
"rings_modules_and_algebras",
"divalg",
"countalg",
"poly",
"orderedzmod",
"numdomain",
"numfield",
"Order.TTheory",
... | [
"Posz",
"int",
"of_hex_uint",
"of_uint"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
print_int (x : int) : Number.int | :=
match x with
| Posz n => Number.IntDecimal (Decimal.Pos (Nat.to_uint n))
| Negz n => Number.IntDecimal (Decimal.Neg (Nat.to_uint n.+1))
end. | Definition | print_int | algebra | algebra/ssrint.v | [
"HB",
"structures",
"mathcomp",
"ssreflect",
"ssrfun",
"ssrbool",
"eqtype",
"ssrnat",
"choice",
"seq",
"fintype",
"finfun",
"bigop",
"order",
"nmodule",
"rings_modules_and_algebras",
"divalg",
"countalg",
"poly",
"orderedzmod",
"numdomain",
"numfield",
"Order.TTheory",
... | [
"Posz",
"int",
"to_uint"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
natsum_of_int (m : int) : nat + nat | :=
match m with Posz p => inl _ p | Negz n => inr _ n end. | Definition | natsum_of_int | algebra | algebra/ssrint.v | [
"HB",
"structures",
"mathcomp",
"ssreflect",
"ssrfun",
"ssrbool",
"eqtype",
"ssrnat",
"choice",
"seq",
"fintype",
"finfun",
"bigop",
"order",
"nmodule",
"rings_modules_and_algebras",
"divalg",
"countalg",
"poly",
"orderedzmod",
"numdomain",
"numfield",
"Order.TTheory",
... | [
"Posz",
"int",
"nat"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
int_of_natsum (m : nat + nat) | :=
match m with inl p => Posz p | inr n => Negz n end. | Definition | int_of_natsum | algebra | algebra/ssrint.v | [
"HB",
"structures",
"mathcomp",
"ssreflect",
"ssrfun",
"ssrbool",
"eqtype",
"ssrnat",
"choice",
"seq",
"fintype",
"finfun",
"bigop",
"order",
"nmodule",
"rings_modules_and_algebras",
"divalg",
"countalg",
"poly",
"orderedzmod",
"numdomain",
"numfield",
"Order.TTheory",
... | [
"Posz",
"nat"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
natsum_of_intK : cancel natsum_of_int int_of_natsum. | Proof. by case. Qed. | Lemma | natsum_of_intK | algebra | algebra/ssrint.v | [
"HB",
"structures",
"mathcomp",
"ssreflect",
"ssrfun",
"ssrbool",
"eqtype",
"ssrnat",
"choice",
"seq",
"fintype",
"finfun",
"bigop",
"order",
"nmodule",
"rings_modules_and_algebras",
"divalg",
"countalg",
"poly",
"orderedzmod",
"numdomain",
"numfield",
"Order.TTheory",
... | [
"int_of_natsum",
"natsum_of_int"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
eqz_nat (m n : nat) : (m%:Z == n%:Z) = (m == n). | Proof. by []. Qed. | Lemma | eqz_nat | algebra | algebra/ssrint.v | [
"HB",
"structures",
"mathcomp",
"ssreflect",
"ssrfun",
"ssrbool",
"eqtype",
"ssrnat",
"choice",
"seq",
"fintype",
"finfun",
"bigop",
"order",
"nmodule",
"rings_modules_and_algebras",
"divalg",
"countalg",
"poly",
"orderedzmod",
"numdomain",
"numfield",
"Order.TTheory",
... | [
"nat"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
addz (m n : int) | :=
match m, n with
| Posz m', Posz n' => Posz (m' + n')
| Negz m', Negz n' => Negz (m' + n').+1
| Posz m', Negz n' => if n' < m' then Posz (m' - n'.+1) else Negz (n' - m')
| Negz n', Posz m' => if n' < m' then Posz (m' - n'.+1) else Negz (n' - m')
end. | Definition | addz | algebra | algebra/ssrint.v | [
"HB",
"structures",
"mathcomp",
"ssreflect",
"ssrfun",
"ssrbool",
"eqtype",
"ssrnat",
"choice",
"seq",
"fintype",
"finfun",
"bigop",
"order",
"nmodule",
"rings_modules_and_algebras",
"divalg",
"countalg",
"poly",
"orderedzmod",
"numdomain",
"numfield",
"Order.TTheory",
... | [
"Posz",
"int",
"n'"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
oppz m | :=
match m with
| Posz n => if n is (n'.+1)%N then Negz n' else Posz 0
| Negz n => Posz (n.+1)%N
end. | Definition | oppz | algebra | algebra/ssrint.v | [
"HB",
"structures",
"mathcomp",
"ssreflect",
"ssrfun",
"ssrbool",
"eqtype",
"ssrnat",
"choice",
"seq",
"fintype",
"finfun",
"bigop",
"order",
"nmodule",
"rings_modules_and_algebras",
"divalg",
"countalg",
"poly",
"orderedzmod",
"numdomain",
"numfield",
"Order.TTheory",
... | [
"Posz",
"n'"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
"-%Z" | := (@oppz) : int_scope. | Notation | -%Z | algebra | algebra/ssrint.v | [
"HB",
"structures",
"mathcomp",
"ssreflect",
"ssrfun",
"ssrbool",
"eqtype",
"ssrnat",
"choice",
"seq",
"fintype",
"finfun",
"bigop",
"order",
"nmodule",
"rings_modules_and_algebras",
"divalg",
"countalg",
"poly",
"orderedzmod",
"numdomain",
"numfield",
"Order.TTheory",
... | [
"oppz"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
"- x" | := (oppz x) : int_scope. | Notation | - x | algebra | algebra/ssrint.v | [
"HB",
"structures",
"mathcomp",
"ssreflect",
"ssrfun",
"ssrbool",
"eqtype",
"ssrnat",
"choice",
"seq",
"fintype",
"finfun",
"bigop",
"order",
"nmodule",
"rings_modules_and_algebras",
"divalg",
"countalg",
"poly",
"orderedzmod",
"numdomain",
"numfield",
"Order.TTheory",
... | [
"oppz"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
"+%Z" | := (@addz) : int_scope. | Notation | +%Z | algebra | algebra/ssrint.v | [
"HB",
"structures",
"mathcomp",
"ssreflect",
"ssrfun",
"ssrbool",
"eqtype",
"ssrnat",
"choice",
"seq",
"fintype",
"finfun",
"bigop",
"order",
"nmodule",
"rings_modules_and_algebras",
"divalg",
"countalg",
"poly",
"orderedzmod",
"numdomain",
"numfield",
"Order.TTheory",
... | [
"addz"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
"x + y" | := (addz x y) : int_scope. | Notation | x + y | algebra | algebra/ssrint.v | [
"HB",
"structures",
"mathcomp",
"ssreflect",
"ssrfun",
"ssrbool",
"eqtype",
"ssrnat",
"choice",
"seq",
"fintype",
"finfun",
"bigop",
"order",
"nmodule",
"rings_modules_and_algebras",
"divalg",
"countalg",
"poly",
"orderedzmod",
"numdomain",
"numfield",
"Order.TTheory",
... | [
"addz"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
"x - y" | := (x + - y) : int_scope. | Notation | x - y | algebra | algebra/ssrint.v | [
"HB",
"structures",
"mathcomp",
"ssreflect",
"ssrfun",
"ssrbool",
"eqtype",
"ssrnat",
"choice",
"seq",
"fintype",
"finfun",
"bigop",
"order",
"nmodule",
"rings_modules_and_algebras",
"divalg",
"countalg",
"poly",
"orderedzmod",
"numdomain",
"numfield",
"Order.TTheory",
... | [] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d | |
PoszD : {morph Posz : m n / (m + n)%N >-> m + n}. | Proof. by []. Qed. | Lemma | PoszD | algebra | algebra/ssrint.v | [
"HB",
"structures",
"mathcomp",
"ssreflect",
"ssrfun",
"ssrbool",
"eqtype",
"ssrnat",
"choice",
"seq",
"fintype",
"finfun",
"bigop",
"order",
"nmodule",
"rings_modules_and_algebras",
"divalg",
"countalg",
"poly",
"orderedzmod",
"numdomain",
"numfield",
"Order.TTheory",
... | [
"Posz"
] | https://github.com/math-comp/math-comp | 91d97df9cf3204b4dab84f4e24bc633e84b6473d |
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