No problem. Build the matrix A. There is no need to transpose the A_i submatrices, then transpose the result again. Just learn to use VERTICAL catenation. So
A = [A_1;A_2;...]
etc. That is, learn what the semicolon does, or learn to use the vertcat function.
Now you have a simple homogeneous linear least squares problem, so a zero right hand side. Solve it using SVD. That is, the solution that minimizes the norm you want, AND has norm(h) == 1, is given by an appropriate column (actually, the last column) of the matrix V, as returned by svd.
[~,~,V] = svd(A);
h = V(:,end);
If the matrix A has less than full rank, then there may be multiple vectors h that satisfy the requirement, but you did not ask for uniqueness, or for all possible solutions in that case. You can simply discard the first two arguments, thus U and S as returned from the SVD, as I did here.
