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

A direct method is porposed 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 contined 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 is illustrated by means of several simple examples.