Computation of projections and eigen distribution in half-planes and disks

This paper provides ecient solutions to the problem of computing projections onto invariant subspaces of a nonsingular matrix corresponding to eigenvalues in specified region in the complex plane such as the Inside or the outside of a circle, half planes or horizontal and vertical strips. Additionally, new form of matrix decompositions called "sector factorization" is introduced in which the polar and sign decompositions are special cases. The main results are based on properties of matrices whose squares are the identity matrix. Several power-like methods for computing the proposed factorizations, particularly the sign and polar decompositions are developed.