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 A1: ( n > 0 & M1 is Nilpotent & M2 is Nilpotent & M1 commutes_with M2 & M1 * M2 = 0. K,n ) ; :: thesis: M1 + M2 is Nilpotent
then A2: ( M1 * M1 = 0. K,n & M2 * M2 = 0. K,n & M1 * M2 = M2 * M1 & M1 * M2 = 0. K,n,n ) by Def2, MATRIX_6:def 1;
(M1 + M2) * (M1 + M2) = (((M1 * M1) + (0. K,n)) + (0. K,n)) + (M2 * M2) by A1, MATRIX_6:36
.= ((M1 * M1) + (0. K,n)) + (M2 * M2) by A2, MATRIX_3:6
.= (0. K,n) + (0. K,n) by A2, MATRIX_3:6
.= 0. K,n,n by MATRIX_3:6
.= 0. K,n ;
hence M1 + M2 is Nilpotent by Def2; :: thesis: verum