The Verification Of Eigenvalue Structure Of A Defective Matrix

In the rapid development of science,many industries require the improvement of the accuracy of matrix eigenvalue computations.Since matrix date are obtained from experiments,there are many kinds of matrix elements.An eigenvalue of a matrix is defective if its algebraic multiplicity is greater than geometric multiplicity.If a matrix has a defective eigenvalue,it is called a defective matrix;otherwise,it is called a single matrix.The computation of defective eigenvalues of matrices is very sensitive to data perturbation and rounding error,which makes it a great challenge in numerical computation.When the matrix data are obtained from experiments,it is particularly difficult to compute the defective eigenvalues and sensitivity analysis.In the computation of matrix eigenvalues,if the perturbation of matrix is arbitrary without preserving the multiplicity support.Then there is a certain type of pseudoeigenvalues that are Lipschitz continuous and backward accurate.If the geometric multiplicity of the eigenvalue and the smallest Jordan block size remain constant after perturbation,Then the perturbation is related to the finite error bound of the defective eigenvalue.Since solving the defective eigenvalue is an ill-posed problem,the least square method is used to transform the problem into an well-posed problem.The matrix of defective eigenvalue has been concerned by scholars in the domestic and overseas because it can reflect the needs of engineering practice.In this paper,the Rump interval algorithm and Kantorovich theorem are used to design a reliable verification algorithm for the defective eigenvalues of the matrix.The main research contents are as follows:(1)Using the definition of the multiplicity support of a defective eigenvalue that is introduced by Zeng,we consider the verification about the sensitivity and computation of a real defective eigenvalue of a real matrix.We discuss how to construct a slightly perturbed interval matrix which is guaranteed to have a real eigenvalue with the computed multiplicity support.Furthermore,we also obtain this eigenvalue and an interval matrix.This interval matrix assuringly contains a real matrix whose columns are the linearly independent eigenvectors of the computed eigenvalue.We demonstrate the effectiveness of the algorithm by several examples.(2)Given an n-order matrix withn different eigenvalues,and its approximate eigenvalue,we propose a verification algorithm for constructing an interval matrix near to given matrix and an interval near to approximate eigenvalue.The computed interval matrix is guaranteed to contain a real defective matrix,which has an exact defective eigenvalue of geometric multiplicity q.