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

for M1, M2 being Matrix of n,K st M2 is invertible & M1 is_similar_to M2 holds

M1 ~ is_similar_to M2 ~

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

M1 ~ is_similar_to M2 ~

let M1, M2 be Matrix of n,K; :: thesis: ( M2 is invertible & M1 is_similar_to M2 implies M1 ~ is_similar_to M2 ~ )

assume that

A1: M2 is invertible and

A2: M1 is_similar_to M2 ; :: thesis: M1 ~ is_similar_to M2 ~

consider M4 being Matrix of n,K such that

A3: M4 is invertible and

A4: M1 = ((M4 ~) * M2) * M4 by A2;

A5: ( M4 ~ is invertible & (M4 ~) ~ = M4 ) by A3, MATRIX_6:16;

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

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

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

( M2 * M4 is invertible & (M4 ~) * (M2 ~) = (M2 * M4) ~ ) by A1, A3, MATRIX_6:36;

then ((M4 ~) * (M2 ~)) * M4 = ((M4 ~) * (M2 * M4)) ~ by A5, MATRIX_6:36

.= M1 ~ by A4, A6, A7, MATRIX_3:33 ;

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

for M1, M2 being Matrix of n,K st M2 is invertible & M1 is_similar_to M2 holds

M1 ~ is_similar_to M2 ~

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

M1 ~ is_similar_to M2 ~

let M1, M2 be Matrix of n,K; :: thesis: ( M2 is invertible & M1 is_similar_to M2 implies M1 ~ is_similar_to M2 ~ )

assume that

A1: M2 is invertible and

A2: M1 is_similar_to M2 ; :: thesis: M1 ~ is_similar_to M2 ~

consider M4 being Matrix of n,K such that

A3: M4 is invertible and

A4: M1 = ((M4 ~) * M2) * M4 by A2;

A5: ( M4 ~ is invertible & (M4 ~) ~ = M4 ) by A3, MATRIX_6:16;

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

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

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

( M2 * M4 is invertible & (M4 ~) * (M2 ~) = (M2 * M4) ~ ) by A1, A3, MATRIX_6:36;

then ((M4 ~) * (M2 ~)) * M4 = ((M4 ~) * (M2 * M4)) ~ by A5, MATRIX_6:36

.= M1 ~ by A4, A6, A7, MATRIX_3:33 ;

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