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;

A5: M4 ~ is_reverse_of M4 by A3, MATRIX_6:def 4;

A6: width ((M4 ~) * M2) = n by MATRIX_0:24;

A7: width (M4 ~) = n by MATRIX_0:24;

A8: ( len (M2 * M4) = n & width (((M4 ~) * M2) * M4) = n ) by MATRIX_0:24;

A9: len (M4 ~) = n by MATRIX_0:24;

A10: width M4 = n by MATRIX_0:24;

A11: ( len M2 = n & width M2 = n ) by MATRIX_0:24;

A12: len M4 = n by MATRIX_0:24;

then M1 * M1 = (((M4 ~) * M2) * M4) * ((M4 ~) * (M2 * M4)) by A4, A11, A7, MATRIX_3:33

.= ((((M4 ~) * M2) * M4) * (M4 ~)) * (M2 * M4) by A9, A7, A8, MATRIX_3:33

.= (((M4 ~) * M2) * (M4 * (M4 ~))) * (M2 * M4) by A12, A10, A9, A6, MATRIX_3:33

.= (((M4 ~) * M2) * (1. (K,n))) * (M2 * M4) by A5, MATRIX_6:def 2

.= ((M4 ~) * M2) * (M2 * M4) by MATRIX_3:19

.= (((M4 ~) * M2) * M2) * M4 by A12, A11, A6, MATRIX_3:33

.= ((M4 ~) * (M2 * M2)) * M4 by A11, A7, MATRIX_3:33

.= ((M4 ~) * (1. (K,n))) * M4 by A2

.= (M4 ~) * M4 by MATRIX_3:19

.= 1. (K,n) by A5, MATRIX_6:def 2 ;

hence M1 is Involutory ; :: thesis: verum

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;

A5: M4 ~ is_reverse_of M4 by A3, MATRIX_6:def 4;

A6: width ((M4 ~) * M2) = n by MATRIX_0:24;

A7: width (M4 ~) = n by MATRIX_0:24;

A8: ( len (M2 * M4) = n & width (((M4 ~) * M2) * M4) = n ) by MATRIX_0:24;

A9: len (M4 ~) = n by MATRIX_0:24;

A10: width M4 = n by MATRIX_0:24;

A11: ( len M2 = n & width M2 = n ) by MATRIX_0:24;

A12: len M4 = n by MATRIX_0:24;

then M1 * M1 = (((M4 ~) * M2) * M4) * ((M4 ~) * (M2 * M4)) by A4, A11, A7, MATRIX_3:33

.= ((((M4 ~) * M2) * M4) * (M4 ~)) * (M2 * M4) by A9, A7, A8, MATRIX_3:33

.= (((M4 ~) * M2) * (M4 * (M4 ~))) * (M2 * M4) by A12, A10, A9, A6, MATRIX_3:33

.= (((M4 ~) * M2) * (1. (K,n))) * (M2 * M4) by A5, MATRIX_6:def 2

.= ((M4 ~) * M2) * (M2 * M4) by MATRIX_3:19

.= (((M4 ~) * M2) * M2) * M4 by A12, A11, A6, MATRIX_3:33

.= ((M4 ~) * (M2 * M2)) * M4 by A11, A7, MATRIX_3:33

.= ((M4 ~) * (1. (K,n))) * M4 by A2

.= (M4 ~) * M4 by MATRIX_3:19

.= 1. (K,n) by A5, MATRIX_6:def 2 ;

hence M1 is Involutory ; :: thesis: verum