nX=5;
nY=5;
nZ=5;
A=rand(nX,nY,nZ);
B=rand(nX,nY,nZ);
C=rand(nX,nY,nZ);
MM=[1/2 -1i/sqrt(2) -1/2; 1/sqrt(2) 0 1/sqrt(2); 1/2 1i/sqrt(2) -1/2];
for x=1:nX
    for y=1:nY
        for z=1:nZ
            mytemp = MM\[A(x,y,z) B(x,y,z) C(x,y,z)]';
            D(x,y,z)=mytemp(1);
            E(x,y,z)=mytemp(2);
            F(x,y,z)=mytemp(3);
        end
    end
end
 
## vectorised
 
X = MM\[A(:)'; B(:)'; C(:)']; 
Dp = reshape(X(1,:), nX, nY, nZ); 
Ep = reshape(X(2,:), nX, nY, nZ); 
Fp = reshape(X(3,:), nX, nY, nZ);
 
norm(D(:) - Dp(:))
norm(E(:) - Ep(:))
norm(F(:) - Fp(:))
 

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