# How to generate a cyclic matrix How to generate all 8×8 cyclic matrix $A$ with the first row $(a_0, a_1, \ldots, a_7)$,

the matrix $A\in GL(8, 2)$ and the first row $a_0 + a_1 + \ldots + a_7 \neq 0$

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What is a cyclic matrix?

Why should have the first row the property that the sum of entries is not zero? (It would maybe be enough to have a non zero row vector, if the question becomes clear.)

A cyclic matrix means a circulant matrix. We need the first row the property that the sum of entries is not zero so that this cyclic matrix could be a invertible matrix.

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Here are some possibilities to generate a circulant matrix, having as input its first row a:

def circulantMatrix0( a ):
n = len(a)
R = range(n)
return matrix( [ [ a[ (j-k) % n ] for j in R ] for k in R ] )

def circulantMatrix1( a ):
n = len(a)
R = range(n)
return matrix( [ [ a[j-k] for j in R ] for k in R ] )

def circulantMatrix2( a, field=GF(2) ):
n = len(a)
T = matrix( F, n, n, [1 if k==(j+1)%n else 0
for j in range(n)
for k in range(n) ] )
return sum( [ a[j]*T^j for j in range(n) ] )

def circulantMatrix3( a, field=GF(2) ):
n = len(a)
b = [a[j] for j in range(n)]
b = b+b
return matrix( n, n, [ b[n-j:2*n-j] for j in range(n) ] )

# Tests:

F = GF(2)
a = vector( F, 8, [1,0,0,0,1,0,0,1] )
A = circulantMatrix0(a)
B = circulantMatrix1(a)
C = circulantMatrix2(a)
D = circulantMatrix3(a)


Then we have:

sage: print A
[1 0 0 0 1 0 0 1]
[1 1 0 0 0 1 0 0]
[0 1 1 0 0 0 1 0]
[0 0 1 1 0 0 0 1]
[1 0 0 1 1 0 0 0]
[0 1 0 0 1 1 0 0]
[0 0 1 0 0 1 1 0]
[0 0 0 1 0 0 1 1]
sage: A==B and A==C and A==D
True
sage: A.charpoly().factor()
(x + 1)^8
sage: A.jordan_form()

[1 1 0 0 0 0 0 0]
[0 1 1 0 0 0 0 0]
[0 0 1 1 0 0 0 0]
[0 0 0 1 1 0 0 0]
[0 0 0 0 1 1 0 0]
[0 0 0 0 0 1 1 0]
[0 0 0 0 0 0 1 1]
[0 0 0 0 0 0 0 1]
sage:

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P.S. Sorry, the field is declared as switch in two of the routines, the last one does not need it, please remove, the other one does not use the field, but the global F, please use the field as base field in the definition of the returned matrix.

The question is interesting! one up...