Eigenvalues of matrices with prescribed submatrices (Q2470922)

The author studies the matrix completion problems. Let \(F\) be a field and let \(n,p_1,p_2,p_3\) be positive integers such that \(n=p_1 +p_2+p_3\). Let \[ C=\left[\begin{matrix} C_{1,1} & C_{1,2} & C_{1,3}\\ C_{2,1} & C_{2,2} & C_{2,3}\\ C_{3,1} & C_{3,2}&C_{3,3} \end{matrix}\right]\in F^{n\times n}, \] where the blocks \(C_{i,j}\) are of type \(p_i\times p_j\), \(i,j\in\{1,2,3\}\). The main theorems describe the list of eigenvalues of \(C\), when \(C_{11},C_{12}\), and \(C_{33}\) are prescribed, and the other six with entries vary over \(F\).