# Null space for smoothed aggregation algebraic multigrid

I do not really get the point of null space usage for creating the prolongation operator for smoothed aggregation algebraic multigrid. I know what the null space is per definition and I know that the null space corresponds with the rigid body modes for structural finite-element problems.

But how can I determine the null space and use it for improving the prolongation operator? Is the null space for 3D structural problems not always the same (3 translations along the global axes and 3 rotation around the global axes)?

Some literature which is understandable and does not skip the needed information would be very helpful.

However, you are asking for the null space of 3D linear elasticity. ..Here, the null space consists of the rigid body modes, i.e. six vectors which are concatenated by the vectors for each node. For one node, three vectors $$b_1, b_2, b_3$$ represent the translation, a simple basis are the three unit vectors. The other three vectors represent the rotation around an (arbitrary) origin (usually the origin of the FEM coordinates are used).
$$b_4 = [0, -z, y]$$ $$b_5 = [z, 0, -x]$$ $$b_6 = [-y, x, 0]$$
So for node $$i$$ you need to take its coordinates $$[x_i, y_i, z_i]$$ to compute, e.g. $$b_4 = [0, -z_i, y_i]$$. Afterwards, you need to concatenate all vectors. So, the null-space is not identical for all FEM models but depends on the coordinates of the nodes.