let n be Nat; :: thesis: 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; :: thesis: 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; :: thesis: ( 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 ; :: thesis: 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, Def2;
(M1 + M2) * (M1 + M2) = (((M1 * M1) + (0. K,n)) + (0. K,n)) + (M2 * M2) by A1, A3, A4, MATRIX_6:36
.= ((M1 * M1) + (0. K,n)) + (M2 * M2) by A5, MATRIX_3:6
.= (0. K,n) + (0. K,n) by A6, A5, MATRIX_3:6
.= 0. K,n,n by MATRIX_3:6
.= 0. K,n ;
hence M1 + M2 is Nilpotent by Def2; :: thesis: verum