let n be Nat; :: thesis: for K being Field
for M1, M2 being Matrix of n,K st M1 is_similar_to M2 & M2 is Nilpotent & n > 0 holds
M1 is Nilpotent
let K be Field; :: thesis: for M1, M2 being Matrix of n,K st M1 is_similar_to M2 & M2 is Nilpotent & n > 0 holds
M1 is Nilpotent
let M1, M2 be Matrix of n,K; :: thesis: ( M1 is_similar_to M2 & M2 is Nilpotent & n > 0 implies M1 is Nilpotent )
assume that
A1: M1 is_similar_to M2 and
A2: M2 is Nilpotent and
A3: n > 0 ; :: thesis: M1 is Nilpotent
consider M4 being Matrix of n,K such that
A4: M4 is invertible and
A5: M1 = ((M4 ~ ) * M2) * M4 by A1, Def5;
A6: M4 ~ is_reverse_of M4 by A4, MATRIX_6:def 4;
A7: width M4 = n by MATRIX_1:25;
A8: width (M4 ~ ) = n by MATRIX_1:25;
A9: width ((M4 ~ ) * M2) = n by MATRIX_1:25;
A10: len (M4 ~ ) = n by MATRIX_1:25;
A11: ( len (M2 * M4) = n & width (((M4 ~ ) * M2) * M4) = n ) by MATRIX_1:25;
A12: len M4 = n by MATRIX_1:25;
A13: ( len M2 = n & width M2 = n ) by MATRIX_1:25;
then M1 * M1 = (((M4 ~ ) * M2) * M4) * ((M4 ~ ) * (M2 * M4)) by A5, A12, A8, MATRIX_3:35
.= ((((M4 ~ ) * M2) * M4) * (M4 ~ )) * (M2 * M4) by A10, A8, A11, MATRIX_3:35
.= (((M4 ~ ) * M2) * (M4 * (M4 ~ ))) * (M2 * M4) by A12, A7, A10, A9, MATRIX_3:35
.= (((M4 ~ ) * M2) * (1. K,n)) * (M2 * M4) by A6, MATRIX_6:def 2
.= ((M4 ~ ) * M2) * (M2 * M4) by MATRIX_3:21
.= (((M4 ~ ) * M2) * M2) * M4 by A12, A13, A9, MATRIX_3:35
.= ((M4 ~ ) * (M2 * M2)) * M4 by A13, A8, MATRIX_3:35
.= ((M4 ~ ) * (0. K,n)) * M4 by A2, Def2
.= (0. K,n,n) * M4 by A3, A10, A8, MATRIX_6:2
.= 0. K,n by A12, A7, MATRIX_6:1 ;
hence M1 is Nilpotent by Def2; :: thesis: verum