Communications Toolbox

gfrank

Compute the rank of a matrix over a Galois field

Syntax

rk = gfrank(A);
rk = gfrank(A,p);

Description

rk = gfrank(A) calculates the rank of the matrix A in GF(2).

rk = gfrank(A,p) calculates the rank of the matrix A in GF(p), where p is a prime number.

Algorithm

gfrank uses an algorithm similar to Gaussian elimination.

Examples

In the code below, gfrank says that the matrix A has less than full rank. This conclusion makes sense because the determinant of A is zero mod 2.

A=[1 0 1;
   1 1 0;
   0 1 1];
det_a = det(A); % Ordinary determinant of A
detmod2 = rem(det(A),2); % Determinant mod 2
rank2 = gfrank(A);
disp(['determinant = ',num2str(det_a)])
disp(['determinant mod 2 is ',num2str(detmod2)])
disp(['rank over GF(2) is ',num2str(rank2)])

The output is below.

determinant = 2
determinant mod 2 is 0
rank over GF(2) is 2

Notice that gflineq finds only the trivial solution to the equation Ax = 0, even though the output above implies that there are infinitely many other solutions.

sol = gflineq(A,[0;0;0])'

sol =

     0     0     0

See Also
gflineq

gfprimfd

gfrepcov