Symmetric tridiagonal partial quadratic inverse eigenvalue problem (SPQIEP)

This paper concerned with the study of the symmetric tridiagonal partial quadratic inverse eigenvalue problem (SPQIEP) for the quadratic matrix pencil. SPQIEP deals with finding the symmetric tridiagonal matrices K ϵ R
^n×n
and C ϵ R
^n×n
from given m (where 1 ≤ m ≤ 2n) eigenpairs so that the corresponding quadratic matrix pencil P(λ) = λ
²
I + λC + K has the given eigenpairs as its eigenvalues and eigenvectors. A necessary and sufficient condition for the existence of solution of this problem is established in this paper. Analytical expressions for the solution K and C of the SPQIEP are presented using the vectorization of a matrix. Main attractive feature of our approach is that our approach does not impose any restrictions on how many eigenpairs should be given for the existence of the solution of this problem. Our proposed approach is applied on different numerical examples to illustrate the validity of our proposed approach.