let n be Element of NAT ; :: thesis: for K being Field
for a being Element of K
for M1 being Matrix of n,K st M1 is col_circulant holds
a * M1 is col_circulant
let K be Field; :: thesis: for a being Element of K
for M1 being Matrix of n,K st M1 is col_circulant holds
a * M1 is col_circulant
let a be Element of K; :: thesis: for M1 being Matrix of n,K st M1 is col_circulant holds
a * M1 is col_circulant
let M1 be Matrix of n,K; :: thesis: ( M1 is col_circulant implies a * M1 is col_circulant )
A1: Indices M1 = [:(Seg n),(Seg n):] by MATRIX_0:24;
assume M1 is col_circulant ; :: thesis: a * M1 is col_circulant
then consider p being FinSequence of K such that
A2: len p = len M1 and
A3: M1 is_col_circulant_about p ;
A4: len M1 = n by MATRIX_0:24;
then A5: dom p = Seg n by A2, FINSEQ_1:def 3;
A6: len (a * p) = len p by MATRIXR1:16;
A7: dom (a * p) = Seg (len (a * p)) by FINSEQ_1:def 3;
A8: for i, j being Nat st [i,j] in Indices (a * M1) holds
(a * M1) * (i,j) = (a * p) . (((i - j) mod (len p)) + 1)

proof

let i, j be Nat; :: thesis: ( [i,j] in Indices (a * M1) implies (a * M1) * (i,j) = (a * p) . (((i - j) mod (len p)) + 1) )
assume [i,j] in Indices (a * M1) ; :: thesis: (a * M1) * (i,j) = (a * p) . (((i - j) mod (len p)) + 1)
then A9: [i,j] in Indices M1 by MATRIX_0:26;
then A10: ((i - j) mod (len p)) + 1 in Seg n by A2, A1, A4, Lm3;
(a * M1) * (i,j) = a * (M1 * (i,j)) by A9, MATRIX_3:def 5
.= (a multfield) . (M1 * (i,j)) by FVSUM_1:49
.= (a multfield) . (p . (((i - j) mod (len p)) + 1)) by A3, A9
.= (a multfield) . (p /. (((i - j) mod (len p)) + 1)) by A5, A10, PARTFUN1:def 6
.= a * (p /. (((i - j) mod (len p)) + 1)) by FVSUM_1:49
.= (a * p) /. (((i - j) mod (len p)) + 1) by A5, A10, POLYNOM1:def 1
.= (a * p) . (((i - j) mod (len p)) + 1) by A2, A4, A6, A7, A10, PARTFUN1:def 6 ;
hence (a * M1) * (i,j) = (a * p) . (((i - j) mod (len p)) + 1) ; :: thesis: verum

end;

A11: len (a * M1) = n by MATRIX_0:24;
len p = n by A2, MATRIX_0:24;
then a * M1 is_col_circulant_about a * p by A11, A6, A8;
then consider q being FinSequence of K such that
A12: ( len q = len (a * M1) & a * M1 is_col_circulant_about q ) ;
take q ; :: according to MATRIX16:def 5 :: thesis: ( len q = len (a * M1) & a * M1 is_col_circulant_about q )
thus ( len q = len (a * M1) & a * M1 is_col_circulant_about q ) by A12; :: thesis: verum