Approximate method for eigensensitivity analysis of a defective matrix

Based on the exact modal expansion method, an arbitrary high-order approximate method is developed for calculating the second-order eigenvalue derivatives and the first-order eigenvector derivatives of a defective matrix. The numerical example shows the validity of the method. If the different eigenvalues μ ( 1 ) , … , μ ( q ) of the matrix are arranged so that | μ ( 1 ) | ≤ ⋯ ≤ | μ ( q ) | and satisfy the condition that | μ ( q 1 ) | < | μ ( q 1 + 1 ) | for some q 1 < q , and if the approximate method only uses the left and right principal eigenvectors associated with μ ( 1 ) , … , μ ( q 1 ) , then associated with μ ( h ) ( h ≤ q 1 ) the errors of the eigenvalue and eigenvector derivatives by the p th-order approximate method are nearly proportional to | μ ( h ) / μ ( q 1 + 1 ) | p + 1 .