linearalgebraingaloisfields.m

To invert a square Galois array, use the inv function. Related is the det function, which computes t

%To invert a square Galois array, use the inv function. Related is the det function, which computes the determinant of a Galois array. Both inv and det behave like their ordinary MATLAB counterparts, except that they perform computations in the Galois field instead of in the field of complex numbers.
%The code below illustrates matrix inversion and determinant computation. 
m = 4;
randommatrix = gf(randint(4,4,2^m),m);
gfid = gf(eye(4),m);
if det(randommatrix) ~= 0
    invmatrix = inv(randommatrix);
    check1 = invmatrix * randommatrix;
    check2 = randommatrix * invmatrix;  
    if (isequal(check1,gfid) & isequal(check2,gfid))
        disp('inv found the correct matrix inverse.');
    end
else
    disp('The matrix is not invertible.');
end
%The output from this example is either of these two messages, depending on whether the randomly generated matrix is nonsingular or singular. inv found the correct matrix inverse.
%The matrix is not invertible.
%Computing RanksTo compute the rank of a Galois array, use the rank function. It behaves like the ordinary MATLAB rank function when given exactly one input argument. The example below illustrates how to find the rank of square and nonsquare Galois arrays.
m = 3;
asquare = gf([4 7 6; 4 6 5; 0 6 1],m);
r1 = rank(asquare);
anonsquare = gf([4 7 6 3; 4 6 5 1; 0 6 1 1],m);
r2 = rank(anonsquare);
[r1 r2]

%Factoring Square MatricesTo express a square Galois array (or a permutation of it) as the product of a lower triangular Galois array and an upper triangular Galois array, use the lu function. This function accepts one input argument and produces exactly two or three output arguments. It behaves like the ordinary MATLAB lu function when given the same syntax. The example below illustrates how to factor using lu. 
tofactor = gf([6 5 7 6; 5 6 2 5; 0 1 7 7; 1 0 5 1],3);
[L,U]=lu(tofactor); % lu with two output arguments
c1 = isequal(L*U, tofactor) % True
tofactor2 = gf([1 2 3 4;1 2 3 0;2 5 2 1; 0 5 0 0],3);
[L2,U2,P] = lu(tofactor2); % lu with three output arguments
c2 = isequal(L2*U2, P*tofactor2) % True