let n be Nat; for K being Field
for M1, M2 being Matrix of n,K st M1 is invertible & M1 commutes_with M2 holds
M1 ~ commutes_with M2
let K be Field; for M1, M2 being Matrix of n,K st M1 is invertible & M1 commutes_with M2 holds
M1 ~ commutes_with M2
let M1, M2 be Matrix of n,K; ( M1 is invertible & M1 commutes_with M2 implies M1 ~ commutes_with M2 )
assume that
A1:
M1 is invertible
and
A2:
M1 commutes_with M2
; M1 ~ commutes_with M2
A3:
M1 ~ is_reverse_of M1
by A1, Def4;
A4:
width M2 = n
by MATRIX_0:24;
A5:
( width M1 = n & len M1 = n )
by MATRIX_0:24;
A6:
( len (M2 * M1) = n & width (M2 * M1) = n )
by MATRIX_0:24;
A7:
len (M1 ~) = n
by MATRIX_0:24;
A8:
len M2 = n
by MATRIX_0:24;
A9:
width (M1 ~) = n
by MATRIX_0:24;
M2 =
(1. (K,n)) * M2
by MATRIX_3:18
.=
((M1 ~) * M1) * M2
by A3
.=
(M1 ~) * (M1 * M2)
by A5, A8, A9, MATRIX_3:33
.=
(M1 ~) * (M2 * M1)
by A2
;
then M2 * (M1 ~) =
(M1 ~) * ((M2 * M1) * (M1 ~))
by A9, A7, A6, MATRIX_3:33
.=
(M1 ~) * (M2 * (M1 * (M1 ~)))
by A5, A4, A7, MATRIX_3:33
.=
(M1 ~) * (M2 * (1. (K,n)))
by A3
.=
(M1 ~) * M2
by MATRIX_3:19
;
hence
M1 ~ commutes_with M2
; verum