let n be Nat; :: thesis: for K being Field
for M1, M2 being Matrix of n,K st 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 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: ( M1 is Nilpotent & M2 is Nilpotent & M1 commutes_with M2 & M1 * M2 = 0. (K,n) implies M1 + M2 is Nilpotent )
assume that
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;
(M1 + M2) * (M1 + M2) = (((M1 * M1) + (0. (K,n))) + (0. (K,n))) + (M2 * M2) by 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 ; :: thesis: verum