On the eigenvalues of matrices with given upper triangular part (Q919062)

The main result is the following. Let \(A=[a_{ij}]^ n_{i,j=1}\) be a matrix with prescribed upper triangular part over an infinite field F. Assume that A admits the block partition \(A=[A_{pq}]^ r_{p,q=1}\) where the matrix \(A_{pq}\) is of size \(n_ p\times n_ p\), \(A_{pq}=0\) for \(p<q\) and each of the square matrices \(A_{pp}\) is lower irreducible. Let \(\alpha_ 1,\alpha_ 2,...,\alpha_ n\) be given elements of F (some values might repeat). Then there exists a completion of A for which \(\alpha_ 1,\alpha_ 2,...,\alpha_ n\) are the roots of the characteristic polynomial if and only if there is a permutation \(\sigma\) on \(\{\) 1,2,...,n\(\}\) such that \(\sum^{m_ p}_{i=m_{p- 1}+1}\alpha_{\sigma (i)}=trace A_{pp}\) \((p=1,2,...,r)\), for \(m_ 0=0\) and \(m_ p=m_{p-1}+n_ p=\sum^{p}_{q=1}n_ q\). Additionally a result regarding the minimal spectral radius of all possible completions of a matrix with given upper triangular part is proven.