From 50ae8e1141ee46aa24f6d4ed61def8d34b1ebcd6 Mon Sep 17 00:00:00 2001
From: Cyril Cohen
Date: Tue, 8 Sep 2020 03:46:31 +0200
Subject: stability by commutation

---
 mathcomp/algebra/mxalgebra.v | 24 ++++++++++++++++++++++++
 1 file changed, 24 insertions(+)

(limited to 'mathcomp/algebra/mxalgebra.v')

diff --git a/mathcomp/algebra/mxalgebra.v b/mathcomp/algebra/mxalgebra.v
index adad2fc..7e213b8 100644
--- a/mathcomp/algebra/mxalgebra.v
+++ b/mathcomp/algebra/mxalgebra.v
@@ -2272,6 +2272,30 @@ Qed.
 Lemma stablemx_unit (n : nat) (V f : 'M[F]_n) : V \in unitmx -> stablemx V f.
 Proof. by move=> Vunit; rewrite submx_full ?row_full_unit. Qed.
 
+Section Commutation.
+
+Variable (n : nat).
+Implicit Types (f g : 'M[F]_n).
+
+Lemma comm_mx_stable (f g : 'M[F]_n) : comm_mx f g -> stablemx f g.
+Proof. by move=> comm_fg; rewrite [_ *m _]comm_fg mulmx_sub. Qed.
+
+Lemma comm_mx_stable_ker (f g : 'M[F]_n) :
+  comm_mx f g -> stablemx (kermx f) g.
+Proof.
+move=> comm_fg; apply/sub_kermxP.
+by rewrite -mulmxA -[g *m _]comm_fg mulmxA mulmx_ker mul0mx.
+Qed.
+
+Lemma comm_mx_stable_eigenspace (f g : 'M[F]_n) a :
+  comm_mx f g -> stablemx (eigenspace f a) g.
+Proof.
+move=> cfg; rewrite comm_mx_stable_ker//.
+by apply/comm_mx_sym/comm_mxB => //; apply:comm_mxC.
+Qed.
+
+End Commutation.
+
 End Stability.
 
 Section DirectSums.
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