let n be Nat; :: thesis: for K being Field
for M1, M2 being Matrix of n,K st M1 is_similar_to M2 holds
M1 + (1. (K,n)) is_similar_to M2 + (1. (K,n))
let K be Field; :: thesis: for M1, M2 being Matrix of n,K st M1 is_similar_to M2 holds
M1 + (1. (K,n)) is_similar_to M2 + (1. (K,n))
let M1, M2 be Matrix of n,K; :: thesis: ( M1 is_similar_to M2 implies M1 + (1. (K,n)) is_similar_to M2 + (1. (K,n)) )
assume M1 is_similar_to M2 ; :: thesis: M1 + (1. (K,n)) is_similar_to M2 + (1. (K,n))
then consider M4 being Matrix of n,K such that
A3: M4 is invertible and
A4: M1 = ((M4 ~) * M2) * M4 ;
A5: M4 ~ is_reverse_of M4 by A3, MATRIX_6:def 4;
A6: ( len (1. (K,n)) = n & width (1. (K,n)) = n ) by MATRIX_0:24;
A7: width ((M4 ~) * M2) = n by MATRIX_0:24;
A8: ( len M4 = n & len ((M4 ~) * M2) = n ) by MATRIX_0:24;
take M4 ; :: according to MATRIX_8:def 5 :: thesis: ( M4 is invertible & M1 + (1. (K,n)) = ((M4 ~) * (M2 + (1. (K,n)))) * M4 )
A9: ( len (M4 ~) = n & width (M4 ~) = n ) by MATRIX_0:24;
( len M2 = n & width M2 = n ) by MATRIX_0:24;
then ((M4 ~) * (M2 + (1. (K,n)))) * M4 = (((M4 ~) * M2) + ((M4 ~) * (1. (K,n)))) * M4 by A9, A6, MATRIX_4:62
.= (((M4 ~) * M2) + (M4 ~)) * M4 by MATRIX_3:19
.= (((M4 ~) * M2) * M4) + ((M4 ~) * M4) by A9, A8, A7, MATRIX_4:63
.= M1 + (1. (K,n)) by A4, A5, MATRIX_6:def 2 ;
hence ( M4 is invertible & M1 + (1. (K,n)) = ((M4 ~) * (M2 + (1. (K,n)))) * M4 ) by A3; :: thesis: verum