A generalized inverse eigenvalue problem and $m$-functions

In this manuscript, a generalized inverse eigenvalue problem is considered that involves a linear pencil $(z\mathcal{J}_{[0,n]}-\mathcal{H}_{[0,n]})$ of matrices arising in the theory of rational interpolation and biorthogonal rational functions. In addition to the reconstruction of the Hermitian matrix $\mathcal{H}_{[0,n]}$ with the entries $b_j's$, characterizations of the rational functions that are components of the prescribed eigenvectors are given. A condition concerning the positive-definiteness of $\mathcal{J}_{[0,n]}$ and which is often an assumption in the direct problem is also isolated. Further, the reconstruction of $\mathcal{H}_{[0,n]}$ is viewed through the inverse of the pencil $(z\mathcal{J}_{[0,n]}-\mathcal{H}_{[0,n]})$ which involves the concept of $m$-functions.