  We investigate numerically efficient approximations of eigenspaces associated
to symmetric and general matrices. The eigenspaces are factored into a fixed
number of fundamental components that can be efficiently manipulated (we
consider extended orthogonal Givens or scaling and shear transformations). The
number of these components controls the trade-off between approximation
accuracy and the computational complexity of projecting on the eigenspaces. We
write minimization problems for the single fundamental components and provide
closed-form solutions. Then we propose algorithms that iterative update all
these components until convergence. We show results on random matrices and an
application on the approximation of graph Fourier transforms for directed and
undirected graphs.
