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 |
|---|---|---|---|---|---|---|
conjmx0(m n : nat) (V : 'M[F]_(m, n)) : conjmx V 0 = 0.
Proof. by rewrite /conjmx mulmx0 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 | conjmx0 | |
conjumx(n : nat) (V : 'M_n) (f : 'M[F]_n) : V \in unitmx ->
conjmx V f = V *m f *m invmx V.
Proof. by move=> uV; rewrite /conjmx pinvmxE. 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 | conjumx | |
conj1mx(n : nat) (f : 'M[F]_n) : conjmx 1%:M f = f.
Proof. by rewrite conjumx ?unitmx1// invmx1 mulmx1 mul1mx. 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 | conj1mx | |
conjVmx(n : nat) (V : 'M_n) (f : 'M[F]_n) : V \in unitmx ->
conjmx (invmx V) f = invmx V *m f *m V.
Proof. by move=> Vunit; rewrite conjumx ?invmxK ?unitmx_inv. 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 | conjVmx | |
conjmxK(n : nat) (V f : 'M[F]_n) :
V \in unitmx -> conjmx (invmx V) (conjmx V f) = f.
Proof. by move=> Vu; rewrite -conjuMumx ?unitmx_inv// mulVmx ?conj1mx. 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 | conjmxK | |
conjmxVK(n : nat) (V f : 'M[F]_n) :
V \in unitmx -> conjmx V (conjmx (invmx V) f) = f.
Proof. by move=> Vu; rewrite -conjuMumx ?unitmx_inv// mulmxV ?conj1mx. 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 | conjmxVK | |
horner_mx_conjm n p (V : 'M[F]_(n.+1, m.+1)) (f : 'M_m.+1) :
row_free V -> stablemx V f ->
horner_mx (conjmx V f) p = conjmx V (horner_mx f p).
Proof.
move=> V_free V_stab; rewrite/conjmx; elim/poly_ind: p => [|p c].
by rewrite !rmorph0 mulmx0 mul0mx.
rewrite !(rmorphD, rmorphM)/= !(horner_mx_X, horner_mx_C) =>... | 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_conj | |
horner_mx_uconjn p (V : 'M[F]_(n.+1)) (f : 'M_n.+1) :
V \is a GRing.unit ->
horner_mx (V *m f *m invmx V) p = V *m horner_mx f p *m invmx V.
Proof.
move=> V_unit; rewrite -!conjumx//.
by rewrite horner_mx_conj ?row_free_unit ?stablemx_unit.
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_uconj | |
horner_mx_uconjCn p (V : 'M[F]_(n.+1)) (f : 'M_n.+1) :
V \is a GRing.unit ->
horner_mx (invmx V *m f *m V) p = invmx V *m horner_mx f p *m V.
Proof.
move=> V_unit; rewrite -[X in _ *m X](invmxK V).
by rewrite horner_mx_uconj ?invmxK ?unitmx_inv.
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_uconjC | |
mxminpoly_conjm n (V : 'M[F]_(m.+1, n.+1)) (f : 'M_n.+1) :
row_free V -> stablemx V f -> mxminpoly (conjmx V f) %| mxminpoly f.
Proof.
by move=> *; rewrite mxminpoly_min// horner_mx_conj// mx_root_minpoly conjmx0.
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_conj | |
mxminpoly_uconjn (V : 'M[F]_(n.+1)) (f : 'M_n.+1) :
V \in unitmx -> mxminpoly (conjmx V f) = mxminpoly f.
Proof.
have simp := (row_free_unit, stablemx_unit, unitmx_inv, unitmx1).
move=> Vu; apply/eqP; rewrite -eqp_monic ?mxminpoly_monic// /eqp.
apply/andP; split; first by rewrite mxminpoly_conj ?simp.
by rewrite -[f ... | 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_uconj | |
sub_kermxpoly_conjmxV f p W : stablemx V f -> row_free V ->
(W <= kermxpoly (conjmx V f) p)%MS = (W *m V <= kermxpoly f p)%MS.
Proof.
case: n m => [|n'] [|m'] in V f W * => fV rfV; rewrite ?thinmx0//.
by rewrite /row_free mxrank.unlock in rfV.
by rewrite mul0mx !sub0mx.
apply/sub_kermxP/sub_kermxP; rewrite horner... | 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 | sub_kermxpoly_conjmx | |
sub_eigenspace_conjmxV f a W : stablemx V f -> row_free V ->
(W <= eigenspace (conjmx V f) a)%MS = (W *m V <= eigenspace f a)%MS.
Proof. by move=> fV rfV; rewrite !eigenspace_poly sub_kermxpoly_conjmx. 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 | sub_eigenspace_conjmx | |
eigenpoly_conjmxV f : stablemx V f -> row_free V ->
{subset eigenpoly (conjmx V f) <= eigenpoly f}.
Proof.
move=> fV rfV a /eigenpolyP [x]; rewrite sub_kermxpoly_conjmx//.
move=> xV_le_fa x_neq0; apply/eigenpolyP.
by exists (x *m V); rewrite ?mulmx_free_eq0.
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 | eigenpoly_conjmx | |
eigenvalue_conjmxV f : stablemx V f -> row_free V ->
{subset eigenvalue (conjmx V f) <= eigenvalue f}.
Proof.
by move=> fV rfV a; rewrite ![_ \in _]eigenvalue_poly; apply: eigenpoly_conjmx.
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 | eigenvalue_conjmx | |
conjmx_eigenvaluea V f : (V <= eigenspace f a)%MS -> row_free V ->
conjmx V f = a%:M.
Proof.
by move=> /eigenspaceP Vfa rfV; rewrite /conjmx Vfa -mul_scalar_mx mulmxKp.
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 | conjmx_eigenvalue | |
restrictmxV := (conjmx (row_base V)). | Notation | 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 | restrictmx | |
simmx_to_for{F : fieldType} {m n}
(P : 'M_(m, n)) A (S : {pred 'M[F]_m}) := S (conjmx P 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 | simmx_to_for | |
simmx_forP A B := (A ~_P {in PredOfSimpl.coerce (pred1 B)}). | Notation | 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 | simmx_for | |
simmx_inS A B := (exists2 P, P \in S & A ~_P B). | Notation | 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 | simmx_in | |
simmx_in_toS A S' := (exists2 P, P \in S & A ~_P {in S'}). | Notation | 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 | simmx_in_to | |
all_simmx_inS As S' :=
(exists2 P, P \in S & all [pred A | A ~_P {in S'}] As). | Notation | 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 | all_simmx_in | |
diagonalizable_forP A := (A ~_P {in is_diag_mx}). | Notation | 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 | diagonalizable_for | |
diagonalizable_inS A := (A ~_{in S} {in is_diag_mx}). | Notation | 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 | diagonalizable_in | |
diagonalizableA := (diagonalizable_in unitmx A). | Notation | 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 | diagonalizable | |
codiagonalizable_inS As := (all_simmx_in S As is_diag_mx). | Notation | 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 | codiagonalizable_in | |
codiagonalizableAs := (codiagonalizable_in unitmx As). | Notation | 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 | codiagonalizable | |
simmxPpm n {P : 'M[F]_(m, n)} {A B} :
stablemx P A -> A ~_P B -> P *m A = B *m P.
Proof. by move=> stablemxPA /eqP <-; rewrite mulmxKpV. 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 | simmxPp | |
simmxWm n {P : 'M[F]_(m, n)} {A B} : row_free P ->
P *m A = B *m P -> A ~_P B.
Proof. by rewrite /(_ ~__ _)/= /conjmx => fP ->; rewrite mulmxKp. 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 | simmxW | |
simmxP{P A B} : P \in unitmx ->
reflect (P *m A = B *m P) (A ~_P B).
Proof.
move=> p_unit; apply: (iffP idP); first exact/simmxPp/stablemx_unit.
by apply: simmxW; rewrite row_free_unit.
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 | simmxP | |
simmxRL{P A B} : P \in unitmx ->
reflect (B = P *m A *m invmx P) (A ~_P B).
Proof. by move=> ?; apply: (iffP eqP); rewrite conjumx. 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 | simmxRL | |
simmxLR{P A B} : P \in unitmx ->
reflect (A = conjmx (invmx P) B) (A ~_P B).
Proof.
by move=> Pu; rewrite conjVmx//; apply: (iffP (simmxRL Pu)) => ->;
rewrite !mulmxA ?(mulmxK, mulmxKV, mulVmx, mulmxV, mul1mx, mulmx1).
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 | simmxLR | |
simmx_minpoly{n} {P A B : 'M[F]_n.+1} : P \in unitmx ->
A ~_P B -> mxminpoly A = mxminpoly B.
Proof. by move=> Pu /eqP<-; rewrite mxminpoly_uconj. 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 | simmx_minpoly | |
diagonalizable_for_row_basem n (P : 'M[F]_(m, n)) (A : 'M_n) :
diagonalizable_for (row_base P) A = is_diag_mx (restrictmx P A).
Proof. by []. 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 | diagonalizable_for_row_base | |
diagonalizable_forPpm n (P : 'M[F]_(m, n)) A :
reflect (forall i j : 'I__, i != j :> nat -> conjmx P A i j = 0)
(diagonalizable_for P A).
Proof. exact: @is_diag_mxP. 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 | diagonalizable_forPp | |
diagonalizable_forPn (P : 'M[F]_n) A : P \in unitmx ->
reflect (forall i j : 'I__, i != j :> nat -> (P *m A *m invmx P) i j = 0)
(diagonalizable_for P A).
Proof. by move=> Pu; rewrite -conjumx//; exact: is_diag_mxP. 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 | diagonalizable_forP | |
diagonalizable_forPex{m} {n} {P : 'M[F]_(m, n)} {A} :
reflect (exists D, A ~_P (diag_mx D)) (diagonalizable_for P A).
Proof. by apply: (iffP (diag_mxP _)) => -[D]/eqP; exists 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 | diagonalizable_forPex | |
diagonalizable_forLRn {P : 'M[F]_n} {A} : P \in unitmx ->
reflect (exists D, A = conjmx (invmx P) (diag_mx D)) (diagonalizable_for P A).
Proof.
by move=> Punit; apply: (iffP diagonalizable_forPex) => -[D /(simmxLR Punit)]; exists 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 | diagonalizable_forLR | |
diagonalizable_for_mxminpoly{n} {P A : 'M[F]_n.+1}
(rs := undup [seq conjmx P A i i | i <- enum 'I_n.+1]) :
P \in unitmx -> diagonalizable_for P A ->
mxminpoly A = \prod_(r <- rs) ('X - r%:P).
Proof.
rewrite /rs => pu /(diagonalizable_forLR pu)[d {A rs}->].
rewrite mxminpoly_uconj ?unitmx_inv// mxminpoly_diag.
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 | diagonalizable_for_mxminpoly | |
diagonalizable_for_sum(F : fieldType) (m n : nat) (p_ : 'I_n -> nat)
(V_ : forall i, 'M[F]_(p_ i, m)) (A : 'M[F]_m) :
mxdirect (\sum_i <<V_ i>>) ->
(forall i, stablemx (V_ i) A) ->
(forall i, row_free (V_ i)) ->
diagonalizable_for (\mxcol_i V_ i) A = [forall i, diagonalizable_for (V_ i) A].
Proof.
m... | 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 | diagonalizable_for_sum | |
codiagonalizable1n (A : 'M[F]_n) :
codiagonalizable [:: A] <-> diagonalizable A.
Proof. by split=> -[P Punit PA]; exists P; move: PA; rewrite //= andbT. 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 | codiagonalizable1 | |
codiagonalizablePfulln (As : seq 'M[F]_n) :
codiagonalizable As
<-> exists m, exists2 P : 'M_(m, n), row_full P &
all [pred A | diagonalizable_for P A] As.
Proof.
split => [[P Punit SPA]|[m [P Pfull SPA]]].
by exists n => //; exists P; rewrite ?row_full_unit.
have Qfull := fullrowsub_unit Pfull.
exists (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 | codiagonalizablePfull | |
codiagonalizable_onm n (V_ : 'I_n -> 'M[F]_m) (As : seq 'M[F]_m) :
(\sum_i V_ i :=: 1%:M)%MS -> mxdirect (\sum_i V_ i) ->
(forall i, all (fun A => stablemx (V_ i) A) As) ->
(forall i, codiagonalizable (map (restrictmx (V_ i)) As)) ->
codiagonalizable As.
Proof.
move=> V1 Vdirect /(_ _)/allP AV /(_ _) /sig... | 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 | codiagonalizable_on | |
diagonalizable_diag{n} (d : 'rV[F]_n) : diagonalizable (diag_mx d).
Proof.
exists 1%:M; rewrite ?unitmx1// /(diagonalizable_for _ _).
by rewrite conj1mx diag_mx_is_diag.
Qed.
Hint Resolve diagonalizable_diag : core. | 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 | diagonalizable_diag | |
diagonalizable_scalar{n} (a : F) : diagonalizable (a%:M : 'M_n).
Proof. by rewrite -diag_const_mx. Qed.
Hint Resolve diagonalizable_scalar : core. | 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 | diagonalizable_scalar | |
diagonalizable0{n} : diagonalizable (0 : 'M[F]_n).
Proof.
by rewrite (_ : 0 = 0%:M)//; apply/matrixP => i j; rewrite !mxE// mul0rn.
Qed.
Hint Resolve diagonalizable0 : core. | 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 | diagonalizable0 | |
diagonalizablePeigen{n} {A : 'M[F]_n} :
diagonalizable A <->
exists2 rs, uniq rs & (\sum_(r <- rs) eigenspace A r :=: 1%:M)%MS.
Proof.
split=> [df|[rs urs rsP]].
suff [rs rsP] : exists rs, (\sum_(r <- rs) eigenspace A r :=: 1%:M)%MS.
exists (undup rs); rewrite ?undup_uniq//; apply: eqmx_trans rsP.
elim: r... | 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 | diagonalizablePeigen | |
diagonalizablePn' (n := n'.+1) (A : 'M[F]_n) :
diagonalizable A <->
exists2 rs, uniq rs & mxminpoly A %| \prod_(x <- rs) ('X - x%:P).
Proof.
split=> [[P Punit /diagonalizable_forPex[d /(simmxLR Punit)->]]|].
rewrite mxminpoly_uconj ?unitmx_inv// mxminpoly_diag.
by eexists; [|by []]; rewrite undup_uniq.
move=> +... | 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 | diagonalizableP | |
diagonalizable_conj_diagm n (V : 'M[F]_(m, n)) (d : 'rV[F]_n) :
stablemx V (diag_mx d) -> row_free V -> diagonalizable (conjmx V (diag_mx d)).
Proof.
case: m n => [|m] [|n] in V d * => Vd rdV; rewrite ?thinmx0.
- by [].
- by [].
- by exfalso; move: rdV; rewrite /row_free mxrank.unlock eqxx orbT.
apply/diagonalizableP... | 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 | diagonalizable_conj_diag | |
codiagonalizablePn (As : seq 'M[F]_n) :
{in As &, forall A B, comm_mx A B} /\ {in As, forall A, diagonalizable A}
<-> codiagonalizable As.
Proof.
split => [cdAs|[P Punit /allP/= AsD]]/=; last first.
split; last by exists P; rewrite // AsD.
move=> A B AAs BAs; move=> /(_ _ _)/diagonalizable_forPex/sigW in AsD.
... | 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 | codiagonalizableP | |
conjmx(m n : nat)
(V : 'M_(m, n)) (f : 'M[F]_n) : 'M_m := V *m f *m pinvmx V. | Definition | algebra | [
"From mathcomp Require Import ssreflect ssrfun ssrbool eqtype ssrnat seq div",
"From mathcomp Require Import choice fintype finfun bigop fingroup perm order",
"From mathcomp Require Import ssralg zmodp matrix mxalgebra poly polydiv mxpoly"
] | algebra/mxred.v | conjmx | |
restrictmxV := (conjmx (row_base V)). | Notation | algebra | [
"From mathcomp Require Import ssreflect ssrfun ssrbool eqtype ssrnat seq div",
"From mathcomp Require Import choice fintype finfun bigop fingroup perm order",
"From mathcomp Require Import ssralg zmodp matrix mxalgebra poly polydiv mxpoly"
] | algebra/mxred.v | restrictmx | |
stablemx_comp(m n p : nat) (V : 'M[F]_(m, n)) (W : 'M_(n, p)) (f : 'M_p) :
stablemx W f -> stablemx V (conjmx W f) -> stablemx (V *m W) f.
Proof. by move=> Wf /(submxMr W); rewrite -mulmxA mulmxKpV// mulmxA. Qed. | Lemma | algebra | [
"From mathcomp Require Import ssreflect ssrfun ssrbool eqtype ssrnat seq div",
"From mathcomp Require Import choice fintype finfun bigop fingroup perm order",
"From mathcomp Require Import ssralg zmodp matrix mxalgebra poly polydiv mxpoly"
] | algebra/mxred.v | stablemx_comp | |
stablemx_restrictm n (A : 'M[F]_n) (V : 'M_n) (W : 'M_(m, \rank V)):
stablemx V A -> stablemx W (restrictmx V A) = stablemx (W *m row_base V) A.
Proof.
move=> A_stabV; rewrite mulmxA -[in RHS]mulmxA.
rewrite -(submxMfree _ W (row_base_free V)) mulmxKpV //.
by rewrite mulmx_sub ?stablemx_row_base.
Qed. | Lemma | algebra | [
"From mathcomp Require Import ssreflect ssrfun ssrbool eqtype ssrnat seq div",
"From mathcomp Require Import choice fintype finfun bigop fingroup perm order",
"From mathcomp Require Import ssralg zmodp matrix mxalgebra poly polydiv mxpoly"
] | algebra/mxred.v | stablemx_restrict | |
conjmxM(m n : nat) (V : 'M[F]_(m, n)) :
{in [pred f | stablemx V f] &, {morph conjmx V : f g / f *m g}}.
Proof.
move=> f g; rewrite !inE => Vf Vg /=.
by rewrite /conjmx 2!mulmxA mulmxA mulmxKpV ?stablemx_row_base.
Qed. | Lemma | algebra | [
"From mathcomp Require Import ssreflect ssrfun ssrbool eqtype ssrnat seq div",
"From mathcomp Require Import choice fintype finfun bigop fingroup perm order",
"From mathcomp Require Import ssralg zmodp matrix mxalgebra poly polydiv mxpoly"
] | algebra/mxred.v | conjmxM | |
conjMmx(m n p : nat) (V : 'M[F]_(m, n)) (W : 'M_(n, p)) (f : 'M_p) :
row_free (V *m W) -> stablemx W f -> stablemx V (conjmx W f) ->
conjmx (V *m W) f = conjmx V (conjmx W f).
Proof.
move=> rfVW Wf VWf; apply: (row_free_inj rfVW); rewrite mulmxKpV ?stablemx_comp//.
by rewrite mulmxA mulmxKpV// -[RHS]mulmxA mulmxKpV... | Lemma | algebra | [
"From mathcomp Require Import ssreflect ssrfun ssrbool eqtype ssrnat seq div",
"From mathcomp Require Import choice fintype finfun bigop fingroup perm order",
"From mathcomp Require Import ssralg zmodp matrix mxalgebra poly polydiv mxpoly"
] | algebra/mxred.v | conjMmx | |
conjuMmx(m n : nat) (V : 'M[F]_m) (W : 'M_(m, n)) (f : 'M_n) :
V \in unitmx -> row_free W -> stablemx W f ->
conjmx (V *m W) f = conjmx V (conjmx W f).
Proof.
move=> Vu rfW Wf; rewrite conjMmx ?stablemx_unit//.
by rewrite /row_free mxrankMfree// -/(row_free V) row_free_unit.
Qed. | Lemma | algebra | [
"From mathcomp Require Import ssreflect ssrfun ssrbool eqtype ssrnat seq div",
"From mathcomp Require Import choice fintype finfun bigop fingroup perm order",
"From mathcomp Require Import ssralg zmodp matrix mxalgebra poly polydiv mxpoly"
] | algebra/mxred.v | conjuMmx | |
conjMumx(m n : nat) (V : 'M[F]_(m, n)) (W f : 'M_n) :
W \in unitmx -> row_free V -> stablemx V (conjmx W f) ->
conjmx (V *m W) f = conjmx V (conjmx W f).
Proof.
move=> Wu rfW Wf; rewrite conjMmx ?stablemx_unit//.
by rewrite /row_free mxrankMfree ?row_free_unit.
Qed. | Lemma | algebra | [
"From mathcomp Require Import ssreflect ssrfun ssrbool eqtype ssrnat seq div",
"From mathcomp Require Import choice fintype finfun bigop fingroup perm order",
"From mathcomp Require Import ssralg zmodp matrix mxalgebra poly polydiv mxpoly"
] | algebra/mxred.v | conjMumx | |
conjuMumx(n : nat) (V W f : 'M[F]_n) :
V \in unitmx -> W \in unitmx ->
conjmx (V *m W) f = conjmx V (conjmx W f).
Proof. by move=> Vu Wu; rewrite conjuMmx ?stablemx_unit ?row_free_unit. Qed. | Lemma | algebra | [
"From mathcomp Require Import ssreflect ssrfun ssrbool eqtype ssrnat seq div",
"From mathcomp Require Import choice fintype finfun bigop fingroup perm order",
"From mathcomp Require Import ssralg zmodp matrix mxalgebra poly polydiv mxpoly"
] | algebra/mxred.v | conjuMumx | |
conjmx_scalar(m n : nat) (V : 'M[F]_(m, n)) (a : F) :
row_free V -> conjmx V a%:M = a%:M.
Proof. by move=> rfV; rewrite /conjmx scalar_mxC mulmxKp. Qed. | Lemma | algebra | [
"From mathcomp Require Import ssreflect ssrfun ssrbool eqtype ssrnat seq div",
"From mathcomp Require Import choice fintype finfun bigop fingroup perm order",
"From mathcomp Require Import ssralg zmodp matrix mxalgebra poly polydiv mxpoly"
] | algebra/mxred.v | conjmx_scalar | |
conj0mx(m n : nat) f : conjmx (0 : 'M[F]_(m, n)) f = 0.
Proof. by rewrite /conjmx !mul0mx. Qed. | Lemma | algebra | [
"From mathcomp Require Import ssreflect ssrfun ssrbool eqtype ssrnat seq div",
"From mathcomp Require Import choice fintype finfun bigop fingroup perm order",
"From mathcomp Require Import ssralg zmodp matrix mxalgebra poly polydiv mxpoly"
] | algebra/mxred.v | conj0mx | |
conjmx0(m n : nat) (V : 'M[F]_(m, n)) : conjmx V 0 = 0.
Proof. by rewrite /conjmx mulmx0 mul0mx. Qed. | Lemma | algebra | [
"From mathcomp Require Import ssreflect ssrfun ssrbool eqtype ssrnat seq div",
"From mathcomp Require Import choice fintype finfun bigop fingroup perm order",
"From mathcomp Require Import ssralg zmodp matrix mxalgebra poly polydiv mxpoly"
] | algebra/mxred.v | conjmx0 | |
conjumx(n : nat) (V : 'M_n) (f : 'M[F]_n) : V \in unitmx ->
conjmx V f = V *m f *m invmx V.
Proof. by move=> uV; rewrite /conjmx pinvmxE. Qed. | Lemma | algebra | [
"From mathcomp Require Import ssreflect ssrfun ssrbool eqtype ssrnat seq div",
"From mathcomp Require Import choice fintype finfun bigop fingroup perm order",
"From mathcomp Require Import ssralg zmodp matrix mxalgebra poly polydiv mxpoly"
] | algebra/mxred.v | conjumx | |
conj1mx(n : nat) (f : 'M[F]_n) : conjmx 1%:M f = f.
Proof. by rewrite conjumx ?unitmx1// invmx1 mulmx1 mul1mx. Qed. | Lemma | algebra | [
"From mathcomp Require Import ssreflect ssrfun ssrbool eqtype ssrnat seq div",
"From mathcomp Require Import choice fintype finfun bigop fingroup perm order",
"From mathcomp Require Import ssralg zmodp matrix mxalgebra poly polydiv mxpoly"
] | algebra/mxred.v | conj1mx | |
conjVmx(n : nat) (V : 'M_n) (f : 'M[F]_n) : V \in unitmx ->
conjmx (invmx V) f = invmx V *m f *m V.
Proof. by move=> Vunit; rewrite conjumx ?invmxK ?unitmx_inv. Qed. | Lemma | algebra | [
"From mathcomp Require Import ssreflect ssrfun ssrbool eqtype ssrnat seq div",
"From mathcomp Require Import choice fintype finfun bigop fingroup perm order",
"From mathcomp Require Import ssralg zmodp matrix mxalgebra poly polydiv mxpoly"
] | algebra/mxred.v | conjVmx | |
conjmxK(n : nat) (V f : 'M[F]_n) :
V \in unitmx -> conjmx (invmx V) (conjmx V f) = f.
Proof. by move=> Vu; rewrite -conjuMumx ?unitmx_inv// mulVmx ?conj1mx. Qed. | Lemma | algebra | [
"From mathcomp Require Import ssreflect ssrfun ssrbool eqtype ssrnat seq div",
"From mathcomp Require Import choice fintype finfun bigop fingroup perm order",
"From mathcomp Require Import ssralg zmodp matrix mxalgebra poly polydiv mxpoly"
] | algebra/mxred.v | conjmxK | |
conjmxVK(n : nat) (V f : 'M[F]_n) :
V \in unitmx -> conjmx V (conjmx (invmx V) f) = f.
Proof. by move=> Vu; rewrite -conjuMumx ?unitmx_inv// mulmxV ?conj1mx. Qed. | Lemma | algebra | [
"From mathcomp Require Import ssreflect ssrfun ssrbool eqtype ssrnat seq div",
"From mathcomp Require Import choice fintype finfun bigop fingroup perm order",
"From mathcomp Require Import ssralg zmodp matrix mxalgebra poly polydiv mxpoly"
] | algebra/mxred.v | conjmxVK | |
horner_mx_conjm n p (B : 'M[F]_(n.+1, m.+1)) (f : 'M_m.+1) :
row_free B -> stablemx B f ->
horner_mx (conjmx B f) p = conjmx B (horner_mx f p).
Proof.
move=> B_free B_stab; rewrite/conjmx; elim/poly_ind: p => [|p c].
by rewrite !rmorph0 mulmx0 mul0mx.
rewrite !(rmorphD, rmorphM)/= !(horner_mx_X, horner_mx_C) =>... | Lemma | algebra | [
"From mathcomp Require Import ssreflect ssrfun ssrbool eqtype ssrnat seq div",
"From mathcomp Require Import choice fintype finfun bigop fingroup perm order",
"From mathcomp Require Import ssralg zmodp matrix mxalgebra poly polydiv mxpoly"
] | algebra/mxred.v | horner_mx_conj | |
horner_mx_uconjn p (B : 'M[F]_(n.+1)) (f : 'M_n.+1) :
B \is a GRing.unit ->
horner_mx (B *m f *m invmx B) p = B *m horner_mx f p *m invmx B.
Proof.
move=> B_unit; rewrite -!conjumx//.
by rewrite horner_mx_conj ?row_free_unit ?stablemx_unit.
Qed. | Lemma | algebra | [
"From mathcomp Require Import ssreflect ssrfun ssrbool eqtype ssrnat seq div",
"From mathcomp Require Import choice fintype finfun bigop fingroup perm order",
"From mathcomp Require Import ssralg zmodp matrix mxalgebra poly polydiv mxpoly"
] | algebra/mxred.v | horner_mx_uconj | |
horner_mx_uconjCn p (B : 'M[F]_(n.+1)) (f : 'M_n.+1) :
B \is a GRing.unit ->
horner_mx (invmx B *m f *m B) p = invmx B *m horner_mx f p *m B.
Proof.
move=> B_unit; rewrite -[X in _ *m X](invmxK B).
by rewrite horner_mx_uconj ?invmxK ?unitmx_inv.
Qed. | Lemma | algebra | [
"From mathcomp Require Import ssreflect ssrfun ssrbool eqtype ssrnat seq div",
"From mathcomp Require Import choice fintype finfun bigop fingroup perm order",
"From mathcomp Require Import ssralg zmodp matrix mxalgebra poly polydiv mxpoly"
] | algebra/mxred.v | horner_mx_uconjC | |
mxminpoly_conjm n (V : 'M[F]_(m.+1, n.+1)) (f : 'M_n.+1) :
row_free V -> stablemx V f -> mxminpoly (conjmx V f) %| mxminpoly f.
Proof.
by move=> *; rewrite mxminpoly_min// horner_mx_conj// mx_root_minpoly conjmx0.
Qed. | Lemma | algebra | [
"From mathcomp Require Import ssreflect ssrfun ssrbool eqtype ssrnat seq div",
"From mathcomp Require Import choice fintype finfun bigop fingroup perm order",
"From mathcomp Require Import ssralg zmodp matrix mxalgebra poly polydiv mxpoly"
] | algebra/mxred.v | mxminpoly_conj | |
mxminpoly_uconjn (V : 'M[F]_(n.+1)) (f : 'M_n.+1) :
V \in unitmx -> mxminpoly (conjmx V f) = mxminpoly f.
Proof.
have simp := (row_free_unit, stablemx_unit, unitmx_inv, unitmx1).
move=> Vu; apply/eqP; rewrite -eqp_monic ?mxminpoly_monic// /eqp.
apply/andP; split; first by rewrite mxminpoly_conj ?simp.
by rewrite -[f ... | Lemma | algebra | [
"From mathcomp Require Import ssreflect ssrfun ssrbool eqtype ssrnat seq div",
"From mathcomp Require Import choice fintype finfun bigop fingroup perm order",
"From mathcomp Require Import ssralg zmodp matrix mxalgebra poly polydiv mxpoly"
] | algebra/mxred.v | mxminpoly_uconj | |
sub_kermxpoly_conjmxV f p W : stablemx V f -> row_free V ->
(W <= kermxpoly (conjmx V f) p)%MS = (W *m V <= kermxpoly f p)%MS.
Proof.
move: n m => [|n'] [|m']// in V f W *; rewrite ?thinmx0// => fV rfV.
- by rewrite /row_free mxrank0 in rfV.
- by rewrite mul0mx !sub0mx.
- apply/sub_kermxP/sub_kermxP; rewrite horner_m... | Lemma | algebra | [
"From mathcomp Require Import ssreflect ssrfun ssrbool eqtype ssrnat seq div",
"From mathcomp Require Import choice fintype finfun bigop fingroup perm order",
"From mathcomp Require Import ssralg zmodp matrix mxalgebra poly polydiv mxpoly"
] | algebra/mxred.v | sub_kermxpoly_conjmx | |
sub_eigenspace_conjmxV f a W : stablemx V f -> row_free V ->
(W <= eigenspace (conjmx V f) a)%MS = (W *m V <= eigenspace f a)%MS.
Proof. by move=> fV rfV; rewrite !eigenspace_poly sub_kermxpoly_conjmx. Qed. | Lemma | algebra | [
"From mathcomp Require Import ssreflect ssrfun ssrbool eqtype ssrnat seq div",
"From mathcomp Require Import choice fintype finfun bigop fingroup perm order",
"From mathcomp Require Import ssralg zmodp matrix mxalgebra poly polydiv mxpoly"
] | algebra/mxred.v | sub_eigenspace_conjmx | |
eigenpoly_conjmxV f : stablemx V f -> row_free V ->
{subset eigenpoly (conjmx V f) <= eigenpoly f}.
Proof.
move=> fV rfV a /eigenpolyP [x]; rewrite sub_kermxpoly_conjmx//.
move=> xV_le_fa x_neq0; apply/eigenpolyP.
by exists (x *m V); rewrite ?mulmx_free_eq0.
Qed. | Lemma | algebra | [
"From mathcomp Require Import ssreflect ssrfun ssrbool eqtype ssrnat seq div",
"From mathcomp Require Import choice fintype finfun bigop fingroup perm order",
"From mathcomp Require Import ssralg zmodp matrix mxalgebra poly polydiv mxpoly"
] | algebra/mxred.v | eigenpoly_conjmx | |
eigenvalue_conjmxV f : stablemx V f -> row_free V ->
{subset eigenvalue (conjmx V f) <= eigenvalue f}.
Proof.
by move=> fV rfV a; rewrite ![_ \in _]eigenvalue_poly; apply: eigenpoly_conjmx.
Qed. | Lemma | algebra | [
"From mathcomp Require Import ssreflect ssrfun ssrbool eqtype ssrnat seq div",
"From mathcomp Require Import choice fintype finfun bigop fingroup perm order",
"From mathcomp Require Import ssralg zmodp matrix mxalgebra poly polydiv mxpoly"
] | algebra/mxred.v | eigenvalue_conjmx | |
conjmx_eigenvaluea V f : (V <= eigenspace f a)%MS -> row_free V ->
conjmx V f = a%:M.
Proof.
by move=> /eigenspaceP Vfa rfV; rewrite /conjmx Vfa -mul_scalar_mx mulmxKp.
Qed. | Lemma | algebra | [
"From mathcomp Require Import ssreflect ssrfun ssrbool eqtype ssrnat seq div",
"From mathcomp Require Import choice fintype finfun bigop fingroup perm order",
"From mathcomp Require Import ssralg zmodp matrix mxalgebra poly polydiv mxpoly"
] | algebra/mxred.v | conjmx_eigenvalue | |
restrictmxV := (conjmx (row_base V)). | Notation | algebra | [
"From mathcomp Require Import ssreflect ssrfun ssrbool eqtype ssrnat seq div",
"From mathcomp Require Import choice fintype finfun bigop fingroup perm order",
"From mathcomp Require Import ssralg zmodp matrix mxalgebra poly polydiv mxpoly"
] | algebra/mxred.v | restrictmx | |
similar_to{F : fieldType} {m n} (P : 'M_(m, n)) A
(C : {pred 'M[F]_m}) := C (conjmx P A). | Definition | algebra | [
"From mathcomp Require Import ssreflect ssrfun ssrbool eqtype ssrnat seq div",
"From mathcomp Require Import choice fintype finfun bigop fingroup perm order",
"From mathcomp Require Import ssralg zmodp matrix mxalgebra poly polydiv mxpoly"
] | algebra/mxred.v | similar_to | |
similarP A B := (similar_to P A (PredOfSimpl.coerce (pred1 B))). | Notation | algebra | [
"From mathcomp Require Import ssreflect ssrfun ssrbool eqtype ssrnat seq div",
"From mathcomp Require Import choice fintype finfun bigop fingroup perm order",
"From mathcomp Require Import ssralg zmodp matrix mxalgebra poly polydiv mxpoly"
] | algebra/mxred.v | similar | |
similar_inD A B := (exists2 P, P \in D & similar P A B). | Notation | algebra | [
"From mathcomp Require Import ssreflect ssrfun ssrbool eqtype ssrnat seq div",
"From mathcomp Require Import choice fintype finfun bigop fingroup perm order",
"From mathcomp Require Import ssralg zmodp matrix mxalgebra poly polydiv mxpoly"
] | algebra/mxred.v | similar_in | |
similar_in_toD A C := (exists2 P, P \in D & similar_to P A C). | Notation | algebra | [
"From mathcomp Require Import ssreflect ssrfun ssrbool eqtype ssrnat seq div",
"From mathcomp Require Import choice fintype finfun bigop fingroup perm order",
"From mathcomp Require Import ssralg zmodp matrix mxalgebra poly polydiv mxpoly"
] | algebra/mxred.v | similar_in_to | |
all_similar_toD As C := (exists2 P, P \in D & all [pred A | similar_to P A C] As). | Notation | algebra | [
"From mathcomp Require Import ssreflect ssrfun ssrbool eqtype ssrnat seq div",
"From mathcomp Require Import choice fintype finfun bigop fingroup perm order",
"From mathcomp Require Import ssralg zmodp matrix mxalgebra poly polydiv mxpoly"
] | algebra/mxred.v | all_similar_to | |
similar_diagP A := (similar_to P A is_diag_mx). | Notation | algebra | [
"From mathcomp Require Import ssreflect ssrfun ssrbool eqtype ssrnat seq div",
"From mathcomp Require Import choice fintype finfun bigop fingroup perm order",
"From mathcomp Require Import ssralg zmodp matrix mxalgebra poly polydiv mxpoly"
] | algebra/mxred.v | similar_diag | |
diagonalizable_inD A := (similar_in_to D A is_diag_mx). | Notation | algebra | [
"From mathcomp Require Import ssreflect ssrfun ssrbool eqtype ssrnat seq div",
"From mathcomp Require Import choice fintype finfun bigop fingroup perm order",
"From mathcomp Require Import ssralg zmodp matrix mxalgebra poly polydiv mxpoly"
] | algebra/mxred.v | diagonalizable_in | |
diagonalizableA := (diagonalizable_in unitmx A). | Notation | algebra | [
"From mathcomp Require Import ssreflect ssrfun ssrbool eqtype ssrnat seq div",
"From mathcomp Require Import choice fintype finfun bigop fingroup perm order",
"From mathcomp Require Import ssralg zmodp matrix mxalgebra poly polydiv mxpoly"
] | algebra/mxred.v | diagonalizable | |
codiagonalizable_inD As := (all_similar_to D As is_diag_mx). | Notation | algebra | [
"From mathcomp Require Import ssreflect ssrfun ssrbool eqtype ssrnat seq div",
"From mathcomp Require Import choice fintype finfun bigop fingroup perm order",
"From mathcomp Require Import ssralg zmodp matrix mxalgebra poly polydiv mxpoly"
] | algebra/mxred.v | codiagonalizable_in | |
codiagonalizableAs := (codiagonalizable_in unitmx As). | Notation | algebra | [
"From mathcomp Require Import ssreflect ssrfun ssrbool eqtype ssrnat seq div",
"From mathcomp Require Import choice fintype finfun bigop fingroup perm order",
"From mathcomp Require Import ssralg zmodp matrix mxalgebra poly polydiv mxpoly"
] | algebra/mxred.v | codiagonalizable | |
similar_trigP A := (similar_to P A is_trig_mx). | Notation | algebra | [
"From mathcomp Require Import ssreflect ssrfun ssrbool eqtype ssrnat seq div",
"From mathcomp Require Import choice fintype finfun bigop fingroup perm order",
"From mathcomp Require Import ssralg zmodp matrix mxalgebra poly polydiv mxpoly"
] | algebra/mxred.v | similar_trig | |
trigonalizable_inD A := (similar_in_to D A is_trig_mx). | Notation | algebra | [
"From mathcomp Require Import ssreflect ssrfun ssrbool eqtype ssrnat seq div",
"From mathcomp Require Import choice fintype finfun bigop fingroup perm order",
"From mathcomp Require Import ssralg zmodp matrix mxalgebra poly polydiv mxpoly"
] | algebra/mxred.v | trigonalizable_in | |
trigonalizableA := (trigonalizable_in unitmx A). | Notation | algebra | [
"From mathcomp Require Import ssreflect ssrfun ssrbool eqtype ssrnat seq div",
"From mathcomp Require Import choice fintype finfun bigop fingroup perm order",
"From mathcomp Require Import ssralg zmodp matrix mxalgebra poly polydiv mxpoly"
] | algebra/mxred.v | trigonalizable | |
cotrigonalizable_inD As := (all_similar_to D As is_trig_mx). | Notation | algebra | [
"From mathcomp Require Import ssreflect ssrfun ssrbool eqtype ssrnat seq div",
"From mathcomp Require Import choice fintype finfun bigop fingroup perm order",
"From mathcomp Require Import ssralg zmodp matrix mxalgebra poly polydiv mxpoly"
] | algebra/mxred.v | cotrigonalizable_in | |
cotrigonalizableAs := (cotrigonalizable_in unitmx As). | Notation | algebra | [
"From mathcomp Require Import ssreflect ssrfun ssrbool eqtype ssrnat seq div",
"From mathcomp Require Import choice fintype finfun bigop fingroup perm order",
"From mathcomp Require Import ssralg zmodp matrix mxalgebra poly polydiv mxpoly"
] | algebra/mxred.v | cotrigonalizable | |
similarPpm n {P : 'M[F]_(m, n)} {A B} :
stablemx P A -> similar P A B -> P *m A = B *m P.
Proof. by move=> stablemxPA /eqP <-; rewrite mulmxKpV. Qed. | Lemma | algebra | [
"From mathcomp Require Import ssreflect ssrfun ssrbool eqtype ssrnat seq div",
"From mathcomp Require Import choice fintype finfun bigop fingroup perm order",
"From mathcomp Require Import ssralg zmodp matrix mxalgebra poly polydiv mxpoly"
] | algebra/mxred.v | similarPp | |
similarWm n {P : 'M[F]_(m, n)} {A B} : row_free P ->
P *m A = B *m P -> similar P A B.
Proof. by rewrite /similar_to/= /conjmx => fP ->; rewrite mulmxKp. Qed. | Lemma | algebra | [
"From mathcomp Require Import ssreflect ssrfun ssrbool eqtype ssrnat seq div",
"From mathcomp Require Import choice fintype finfun bigop fingroup perm order",
"From mathcomp Require Import ssralg zmodp matrix mxalgebra poly polydiv mxpoly"
] | algebra/mxred.v | similarW | |
similarP{p f g} : p \in unitmx ->
reflect (p *m f = g *m p) (similar p f g).
Proof.
move=> p_unit; apply: (iffP idP); first exact/similarPp/stablemx_unit.
by apply: similarW; rewrite row_free_unit.
Qed. | Lemma | algebra | [
"From mathcomp Require Import ssreflect ssrfun ssrbool eqtype ssrnat seq div",
"From mathcomp Require Import choice fintype finfun bigop fingroup perm order",
"From mathcomp Require Import ssralg zmodp matrix mxalgebra poly polydiv mxpoly"
] | algebra/mxred.v | similarP | |
similarRL{p f g} : p \in unitmx ->
reflect (g = p *m f *m invmx p) (similar p f g).
Proof. by move=> ?; apply: (iffP eqP); rewrite conjumx. Qed. | Lemma | algebra | [
"From mathcomp Require Import ssreflect ssrfun ssrbool eqtype ssrnat seq div",
"From mathcomp Require Import choice fintype finfun bigop fingroup perm order",
"From mathcomp Require Import ssralg zmodp matrix mxalgebra poly polydiv mxpoly"
] | algebra/mxred.v | similarRL | |
similarLR{p f g} : p \in unitmx ->
reflect (f = conjmx (invmx p) g) (similar p f g).
Proof.
by move=> pu; rewrite conjVmx//; apply: (iffP (similarRL pu)) => ->;
rewrite !mulmxA ?(mulmxK, mulmxKV, mulVmx, mulmxV, mul1mx, mulmx1).
Qed. | Lemma | algebra | [
"From mathcomp Require Import ssreflect ssrfun ssrbool eqtype ssrnat seq div",
"From mathcomp Require Import choice fintype finfun bigop fingroup perm order",
"From mathcomp Require Import ssralg zmodp matrix mxalgebra poly polydiv mxpoly"
] | algebra/mxred.v | similarLR | |
similar_mxminpoly{n} {p f g : 'M[F]_n.+1} : p \in unitmx ->
similar p f g -> mxminpoly f = mxminpoly g.
Proof. by move=> pu /eqP<-; rewrite mxminpoly_uconj. Qed. | Lemma | algebra | [
"From mathcomp Require Import ssreflect ssrfun ssrbool eqtype ssrnat seq div",
"From mathcomp Require Import choice fintype finfun bigop fingroup perm order",
"From mathcomp Require Import ssralg zmodp matrix mxalgebra poly polydiv mxpoly"
] | algebra/mxred.v | similar_mxminpoly | |
similar_diag_row_basem n (P : 'M[F]_(m, n)) (A : 'M_n) :
similar_diag (row_base P) A = is_diag_mx (restrictmx P A).
Proof. by []. Qed. | Lemma | algebra | [
"From mathcomp Require Import ssreflect ssrfun ssrbool eqtype ssrnat seq div",
"From mathcomp Require Import choice fintype finfun bigop fingroup perm order",
"From mathcomp Require Import ssralg zmodp matrix mxalgebra poly polydiv mxpoly"
] | algebra/mxred.v | similar_diag_row_base |
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