Approximate method for eigensensitivity analysis of a defective matrix

Abstract

a b s t r a c tBased 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 pth-order approximate method are nearly proportional to |µ(h)/µ(q 1 + 1)| p+1 .