let n be Nat; :: thesis: for K being Field

for M1 being Matrix of n,K st M1 is Self_Reversible holds

M1 is Involutory

let K be Field; :: thesis: for M1 being Matrix of n,K st M1 is Self_Reversible holds

M1 is Involutory

let M1 be Matrix of n,K; :: thesis: ( M1 is Self_Reversible implies M1 is Involutory )

assume A1: M1 is Self_Reversible ; :: thesis: M1 is Involutory

then M1 is invertible ;

then M1 ~ is_reverse_of M1 by MATRIX_6:def 4;

then A2: M1 * (M1 ~) = 1. (K,n) by MATRIX_6:def 2;

M1 * M1 = M1 * (M1 ~) by A1;

hence M1 is Involutory by A2; :: thesis: verum

for M1 being Matrix of n,K st M1 is Self_Reversible holds

M1 is Involutory

let K be Field; :: thesis: for M1 being Matrix of n,K st M1 is Self_Reversible holds

M1 is Involutory

let M1 be Matrix of n,K; :: thesis: ( M1 is Self_Reversible implies M1 is Involutory )

assume A1: M1 is Self_Reversible ; :: thesis: M1 is Involutory

then M1 is invertible ;

then M1 ~ is_reverse_of M1 by MATRIX_6:def 4;

then A2: M1 * (M1 ~) = 1. (K,n) by MATRIX_6:def 2;

M1 * M1 = M1 * (M1 ~) by A1;

hence M1 is Involutory by A2; :: thesis: verum