Matrix left division \ of Galois arrays
x = A\B
x = A\B divides the Galois array
A into B to produce a particular solution
of the linear equation A*x = B. In the special case when
A is a nonsingular square matrix, x is the
unique solution, inv(A)*B, to the equation.
The code below shows that A \ eye(size(A)) is the
inverse of the nonsingular square matrix A.
m = 4; A = gf([8 1 6; 3 5 7; 4 9 2],m); Id = gf(eye(size(A)),m); X = A \ Id; ck1 = isequal(X*A, Id) ck2 = isequal(A*X, Id)
The output is below.
ck1 =
1
ck2 =
1
Other examples are in Solving Linear Equations.
The matrix A must be one of these types:
A nonsingular square matrix
A matrix, in which there are more rows than columns, such that
A'*A is nonsingular
A matrix, in which there are more columns than rows, such that
A*A' is nonsingular
If A is an M-by-N matrix where M > N,
A \ B is the same as
(A'*A) \ (A'*B).
If A is an M-by-N matrix where M < N,
A \ B is the same as
A' * ((A*A') \ B). This solution is not
unique.