let n be Nat; for K being Field
for M1, M2 being Matrix of n,K st n > 0 & M1 is Nilpotent & M2 is Nilpotent & M1 commutes_with M2 & M1 * M2 = 0. (K,n) holds
M1 + M2 is Nilpotent
let K be Field; for M1, M2 being Matrix of n,K st n > 0 & M1 is Nilpotent & M2 is Nilpotent & M1 commutes_with M2 & M1 * M2 = 0. (K,n) holds
M1 + M2 is Nilpotent
let M1, M2 be Matrix of n,K; ( n > 0 & M1 is Nilpotent & M2 is Nilpotent & M1 commutes_with M2 & M1 * M2 = 0. (K,n) implies M1 + M2 is Nilpotent )
assume that
A1:
n > 0
and
A2:
( M1 is Nilpotent & M2 is Nilpotent )
and
A3:
M1 commutes_with M2
and
A4:
M1 * M2 = 0. (K,n)
; M1 + M2 is Nilpotent
A5:
M1 * M2 = 0. (K,n,n)
by A4;
A6:
( M1 * M1 = 0. (K,n) & M2 * M2 = 0. (K,n) )
by A2;
(M1 + M2) * (M1 + M2) =
(((M1 * M1) + (0. (K,n))) + (0. (K,n))) + (M2 * M2)
by A1, A3, A4, MATRIX_6:35
.=
((M1 * M1) + (0. (K,n))) + (M2 * M2)
by A5, MATRIX_3:4
.=
(0. (K,n)) + (0. (K,n))
by A6, A5, MATRIX_3:4
.=
0. (K,n,n)
by MATRIX_3:4
.=
0. (K,n)
;
hence
M1 + M2 is Nilpotent
; verum