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The examples below illustrate the four division operators in a Galois field by computing multiplicative inverses of individual elements and of an array. You can also compute inverses using inv or using exponentiation by -1.
This example divides 1 by each of the individual elements in a Galois array using the ./ and .\ operators. These two operators differ only in their sequence of input arguments. Each quotient vector lists the multiplicative inverses of the nonzero elements of the field. In this example, MATLAB expands the scalar 1 to the size of nz before computing; alternatively, you can use as arguments two arrays of the same size.
m = 5; nz = gf([1:2^m-1],m); % Nonzero elements of the field inv1 = 1 ./ nz; % Divide 1 by each element. inv2 = nz .\ 1; % Obtain same result using .\ operator.
This example divides the identity array by the square Galois array mat using the / and \ operators. Each quotient matrix is the multiplicative inverse of mat. Notice how the transpose operator (') appears in the equivalent operation using \. For square matrices, the sequence of transpose operations is unnecessary, but for nonsquare matrices, it is necessary.
m = 5; mat = gf([1 2 3; 4 5 6; 7 8 9],m); minv1 = eye(3) / mat; % Compute matrix inverse. minv2 = (mat' \ eye(3)')'; % Obtain same result using \ operator.
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