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 =


ck2 =


Other examples are in Solving Linear Equations.


The matrix A must be one of these types:

  • A nonsingular square matrix

  • A tall matrix such that A'*A is nonsingular

  • A wide matrix such that A*A' is nonsingular

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If A is an M-by-N tall matrix where M > N, A \ B is the same as (A'*A) \ (A'*B).

If A is an M-by-N wide matrix where M < N, A \ B is the same as A' * ((A*A') \ B). This solution is not unique.

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