let n be Element of NAT ; :: thesis: for K being Field
for M1 being Matrix of n,K st M1 is col_circulant holds
- M1 is col_circulant
let K be Field; :: thesis: for M1 being Matrix of n,K st M1 is col_circulant holds
- M1 is col_circulant
let M1 be Matrix of n,K; :: thesis: ( M1 is col_circulant implies - M1 is col_circulant )
A1: Indices M1 = [:(Seg n),(Seg n):] by MATRIX_0:24;
A2: Indices (- M1) = [:(Seg n),(Seg n):] by MATRIX_0:24;
assume M1 is col_circulant ; :: thesis: - M1 is col_circulant
then consider p being FinSequence of K such that
A3: len p = len M1 and
A4: M1 is_col_circulant_about p ;
p is Element of (len p) -tuples_on the carrier of K by FINSEQ_2:92;
then A5: - p is Element of (len p) -tuples_on the carrier of K by FINSEQ_2:113;
then A6: ( len (- M1) = n & len (- p) = len p ) by CARD_1:def 7, MATRIX_0:24;
A7: len M1 = n by MATRIX_0:24;
then A8: dom p = Seg n by A3, FINSEQ_1:def 3;
for i, j being Nat st [i,j] in Indices (- M1) holds
(- M1) * (i,j) = (- p) . (((i - j) mod (len (- p))) + 1)

proof

let i, j be Nat; :: thesis: ( [i,j] in Indices (- M1) implies (- M1) * (i,j) = (- p) . (((i - j) mod (len (- p))) + 1) )
assume A9: [i,j] in Indices (- M1) ; :: thesis: (- M1) * (i,j) = (- p) . (((i - j) mod (len (- p))) + 1)
then A10: ((i - j) mod n) + 1 in Seg n by A2, Lm3;
(- M1) * (i,j) = - (M1 * (i,j)) by A1, A2, A9, MATRIX_3:def 2
.= (comp K) . (M1 * (i,j)) by VECTSP_1:def 13
.= (comp K) . (p . (((i - j) mod (len p)) + 1)) by A4, A1, A2, A9
.= (- p) . (((i - j) mod (len p)) + 1) by A3, A7, A8, A10, FUNCT_1:13 ;
hence (- M1) * (i,j) = (- p) . (((i - j) mod (len (- p))) + 1) by A5, CARD_1:def 7; :: thesis: verum

end;

then - M1 is_col_circulant_about - p by A3, A7, A6;
then consider r being FinSequence of K such that
A11: ( len r = len (- M1) & - M1 is_col_circulant_about r ) ;
take r ; :: according to MATRIX16:def 5 :: thesis: ( len r = len (- M1) & - M1 is_col_circulant_about r )
thus ( len r = len (- M1) & - M1 is_col_circulant_about r ) by A11; :: thesis: verum