On finding eigenvalue distribution of a matrix in several regions of the complex plane

A direct method is proposed for determining eigenvalue distribution of a matrix with respect to several important regions of the complex plane. These regions include half planes, shifted half planes, hyperbolas, sectors, quadrants, imaginary axis, region contained within two straight lines that pass through the orgin, etc. The method neither requires computation of the characteristic polynomial of the given matrix nor solution of any matrix equations. The method seems to be more efficient than the eigenvalue and matrix equations methods.