  In this paper, linearly structured partial polynomial inverse eigenvalue
problem is considered for the $n\times n$ matrix polynomial of arbitrary degree
$k$. Given a set of $m$ eigenpairs ($1 \leqslant m \leqslant kn$), this problem
concerns with computing the matrices $A_i\in{\mathbb{R}^{n\times n}}$ for
$i=0,1,2, \ldots ,(k-1)$ of specified linear structure such that the matrix
polynomial $P(\lambda)=\lambda^k I_n +\sum_{i=0}^{k-1} \lambda^{i} A_{i}$ has
the given eigenpairs as its eigenvalues and eigenvectors. Many practical
applications give rise to the linearly structured structured matrix polynomial.
Therefore, construction of the linearly structured matrix polynomial is the
most important aspect of the polynomial inverse eigenvalue problem(PIEP). In
this paper, a necessary and sufficient condition for the existence of the
solution of this problem is derived. Additionally, we characterize the class of
all solutions to this problem by giving the explicit expressions of solutions.
The results presented in this paper address some important open problems in the
area of PIEP raised in De Teran, Dopico and Van Dooren [SIAM Journal on Matrix
Analysis and Applications, $36(1)$ ($2015$), pp $302-328$]. An attractive
feature of our solution approach is that it does not impose any restriction on
the number of eigendata for computing the solution of PIEP. The proposed method
is validated with various numerical examples on a spring mass problem.
