The Rresearch Of Inhomogeneous Eigenvalue Problems For Matrices

The matrix inhomogeneous eigenvalue problem is widely applied in mathematics and many other fields,for instance,the studies of the stability of solutions of linear differential equations and the stationary values of a second-degree polynomial on the unit sphere,and the constrained eigenvalue problem and so on. We have studied the existence and the numbers of solutions,meanwhile,we have given some numerical methods to compute them.In this paper,first of all,we discussed the properties of the solutions for the inhomogeneous eigenvalue of the matrices which have n different eigenvalues,and the validity is prooved with an example.Secondly,we improved the related inclusion region for it,which is based on the eigenvalue inclusion region theories such as Gerschgorin theory and ovals of Cassini theory. The results have established a theoretical foudation for the appliance and calculation for the matrix inhomogeneous eigenvalue.