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 Involutory holds
M1 is Involutory
let K be Field; :: thesis: for M1, M2 being Matrix of n,K st M1 is_similar_to M2 & M2 is Involutory holds
M1 is Involutory
let M1, M2 be Matrix of n,K; :: thesis: ( M1 is_similar_to M2 & M2 is Involutory implies M1 is Involutory )
assume that
A1: M1 is_similar_to M2 and
A2: M2 is Involutory ; :: thesis: M1 is Involutory
consider M4 being Matrix of n,K such that
A3: M4 is invertible and
A4: M1 = ((M4 ~ ) * M2) * M4 by A1, Def5;
A5: M4 ~ is_reverse_of M4 by A3, MATRIX_6:def 4;
A6: width ((M4 ~ ) * M2) = n by MATRIX_1:25;
A7: width (M4 ~ ) = n by MATRIX_1:25;
A8: ( len (M2 * M4) = n & width (((M4 ~ ) * M2) * M4) = n ) by MATRIX_1:25;
A9: len (M4 ~ ) = n by MATRIX_1:25;
A10: width M4 = n by MATRIX_1:25;
A11: ( len M2 = n & width M2 = n ) by MATRIX_1:25;
A12: len M4 = n by MATRIX_1:25;
then M1 * M1 = (((M4 ~ ) * M2) * M4) * ((M4 ~ ) * (M2 * M4)) by A4, A11, A7, MATRIX_3:35
.= ((((M4 ~ ) * M2) * M4) * (M4 ~ )) * (M2 * M4) by A9, A7, A8, MATRIX_3:35
.= (((M4 ~ ) * M2) * (M4 * (M4 ~ ))) * (M2 * M4) by A12, A10, A9, A6, MATRIX_3:35
.= (((M4 ~ ) * M2) * (1. K,n)) * (M2 * M4) by A5, MATRIX_6:def 2
.= ((M4 ~ ) * M2) * (M2 * M4) by MATRIX_3:21
.= (((M4 ~ ) * M2) * M2) * M4 by A12, A11, A6, MATRIX_3:35
.= ((M4 ~ ) * (M2 * M2)) * M4 by A11, A7, MATRIX_3:35
.= ((M4 ~ ) * (1. K,n)) * M4 by A2, Def3
.= (M4 ~ ) * M4 by MATRIX_3:21
.= 1. K,n by A5, MATRIX_6:def 2 ;
hence M1 is Involutory by Def3; :: thesis: verum