cyclicMatrix -- cyclic matrix

Synopsis

Usage:

cyclicMatrix(v)

cyclicMatrix(F,v)
Inputs:
- v, a list,
- F, a Galois field,
Outputs:
- a matrix, $M$

Description

A cyclic matrix (also known as circulant matrix) is a square matrix whose every row (starting from the second) is the right cyclic shift by one entry of the previous row. Below we present ways to define a cyclic matrix $M$.

a list is given

Usage:

cyclicMatrix(v)
Inputs:
- v, a list,
Outputs:
- a matrix, $M$

Assume that $v_1,\ldots,v_n$ are the entries of v. The output is an $n\times n$ matrix. The entries of row 1 are $v_1,\ldots,v_n$, the entries of row 2 are $v_2,\ldots,v_n,v_1$, in general, the entries of row $i$ are $v_i,\ldots,v_n,v_1,\ldots, v_{i-1}$.

i1 : v=toList(1,0,2,0);

i2 : M =cyclicMatrix(v)

o2 = | 1 0 2 0 |
     | 0 1 0 2 |
     | 2 0 1 0 |
     | 0 2 0 1 |

              4       4
o2 : Matrix ZZ  <-- ZZ

a finite field and a list are given

Usage:

cyclicMatrix(v,F)
Inputs:
- v, a list,
- F, a Galois field,
Outputs:
- a matrix, $M$

Assume that $v_1,\ldots,v_n$ are the entries of v. The output is an $n\times n$ matrix over F. The entries of row 1 are $v_1,\ldots,v_n$, the entries of row 2 are $v_2,\ldots,v_n,v_1$, in general, the entries of row $i$ are $v_i,\ldots,v_n,v_1,\ldots, v_{i-1}$.

i3 : F=GF(4,Variable=>a);

i4 : v={0,1,a};

i5 : M=cyclicMatrix(F,v)

o5 = | 0 1 a |
     | a 0 1 |
     | 1 a 0 |

             3      3
o5 : Matrix F  <-- F

Ways to use cyclicMatrix :

cyclicMatrix(GaloisField,List)
cyclicMatrix(List)

For the programmer

The object cyclicMatrix is a method function.