let n be Nat; for K being Field
for M1, M2 being Matrix of n,K st M1 commutes_with M2 holds
M1 * M1 commutes_with M2
let K be Field; for M1, M2 being Matrix of n,K st M1 commutes_with M2 holds
M1 * M1 commutes_with M2
let M1, M2 be Matrix of n,K; ( M1 commutes_with M2 implies M1 * M1 commutes_with M2 )
A1:
width M2 = n
by MATRIX_0:24;
A2:
( width M1 = n & len M1 = n )
by MATRIX_0:24;
assume A3:
M1 commutes_with M2
; M1 * M1 commutes_with M2
A4:
len M2 = n
by MATRIX_0:24;
then (M1 * M1) * M2 =
M1 * (M1 * M2)
by A2, MATRIX_3:33
.=
M1 * (M2 * M1)
by A3
.=
(M1 * M2) * M1
by A1, A2, A4, MATRIX_3:33
.=
(M2 * M1) * M1
by A3
.=
M2 * (M1 * M1)
by A1, A2, MATRIX_3:33
;
hence
M1 * M1 commutes_with M2
; verum