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

for M1, M2 being Matrix of n,K st M1 is_similar_to M2 & n > 0 holds

M1 + M1 is_similar_to M2 + M2

let K be Field; :: thesis: for M1, M2 being Matrix of n,K st M1 is_similar_to M2 & n > 0 holds

M1 + M1 is_similar_to M2 + M2

let M1, M2 be Matrix of n,K; :: thesis: ( M1 is_similar_to M2 & n > 0 implies M1 + M1 is_similar_to M2 + M2 )

assume that

A1: M1 is_similar_to M2 and

A2: n > 0 ; :: thesis: M1 + M1 is_similar_to M2 + M2

consider M4 being Matrix of n,K such that

A3: M4 is invertible and

A4: M1 = ((M4 ~) * M2) * M4 by A1;

A5: width ((M4 ~) * M2) = n by MATRIX_0:24;

take M4 ; :: according to MATRIX_8:def 5 :: thesis: ( M4 is invertible & M1 + M1 = ((M4 ~) * (M2 + M2)) * M4 )

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

A7: ( len M2 = n & width M2 = n ) by MATRIX_0:24;

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

then ((M4 ~) * (M2 + M2)) * M4 = (((M4 ~) * M2) + ((M4 ~) * M2)) * M4 by A2, A7, MATRIX_4:62

.= M1 + M1 by A2, A4, A6, A5, MATRIX_4:63 ;

hence ( M4 is invertible & M1 + M1 = ((M4 ~) * (M2 + M2)) * M4 ) by A3; :: thesis: verum

for M1, M2 being Matrix of n,K st M1 is_similar_to M2 & n > 0 holds

M1 + M1 is_similar_to M2 + M2

let K be Field; :: thesis: for M1, M2 being Matrix of n,K st M1 is_similar_to M2 & n > 0 holds

M1 + M1 is_similar_to M2 + M2

let M1, M2 be Matrix of n,K; :: thesis: ( M1 is_similar_to M2 & n > 0 implies M1 + M1 is_similar_to M2 + M2 )

assume that

A1: M1 is_similar_to M2 and

A2: n > 0 ; :: thesis: M1 + M1 is_similar_to M2 + M2

consider M4 being Matrix of n,K such that

A3: M4 is invertible and

A4: M1 = ((M4 ~) * M2) * M4 by A1;

A5: width ((M4 ~) * M2) = n by MATRIX_0:24;

take M4 ; :: according to MATRIX_8:def 5 :: thesis: ( M4 is invertible & M1 + M1 = ((M4 ~) * (M2 + M2)) * M4 )

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

A7: ( len M2 = n & width M2 = n ) by MATRIX_0:24;

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

then ((M4 ~) * (M2 + M2)) * M4 = (((M4 ~) * M2) + ((M4 ~) * M2)) * M4 by A2, A7, MATRIX_4:62

.= M1 + M1 by A2, A4, A6, A5, MATRIX_4:63 ;

hence ( M4 is invertible & M1 + M1 = ((M4 ~) * (M2 + M2)) * M4 ) by A3; :: thesis: verum