On a quadratic inverse eigenvalue problem
Yunfeng Cai, Shufang Xu · Inverse Problems · 2009
This paper concerns the quadratic inverse eigenvalue problem (QIEP) of constructing real symmetric matrices M , C and K of size n × n , with M nonsingular, so that the quadratic matrix polynomial Q (λ) ≡ λ 2 M + λ C + K has a completely prescribed set of eigenvalues and eigenvectors. It is shown via construction that the QIEP has a solution if and only if r 0, where r and δ are computable from the prescribed spectral data. A necessary and sufficient condition for the existence of a solution to the QIEP with M being positive definite is also established in a constructive way. Furthermore, two algorithms are developed: one is to solve the QIEP; another is to find a particular solution to the QIEP with the leading coefficient matrix being positive definite, which also provides us an approach to a simultaneous reduction of real symmetric matrix triple ( M , C , K ) by real congruence. Numerical results show that the two algorithms are feasible and numerically reliable.