Solution of the linearly structured partial polynomial inverse eigenvalue problem

In this paper, we consider the linearly structured partial polynomial inverse eigenvalue problem (LPPIEP) of constructing matrices Ai∈Rn×n for i=0,1,2,…,(k−1) specified linear structure such that matrix P(λ)=λkIn+∑i=0k−1λiAi has m (1⩽m⩽kn) prescribed eigenpairs as its eigenvalues and eigenvectors. Many practical applications give rise to polynomials. Typical are symmetric, skew-symmetric, tridiagonal, diagonal, pentagonal, Hankel, Toeplitz, etc. Therefore, construction with aforementioned structures is an important but challenging aspect (PIEP). a necessary sufficient condition existence solution derived. Additionally, characterize class all solutions by giving explicit expressions solutions. It should be emphasized results presented in paper resolve some open problems area PIEP namely, polynomials alternating pointed out De Terán et al. (2015). Further, study sensitivity perturbation eigendata. An attractive feature our approach it does not impose any restriction on number eigendata computing LPPIEP. Towards end, proposed method validated various numerical examples spring mass problem.