Geometric multiplicities of completions of partial triangular matrices (Q1893089)

The author studies the possible geometric multiplicities of \(\lambda_ 1,\dots, \lambda_ n\) eigenvalues of a completed matrix. It is for convenience to consider the completed matrix \(A+ T\) as a strictly lower triangular additive perturbation of a full \(n\)-by-\(n\) complex matrix \(A\). The following is the main result: Theorem 1. Given an \(n\times n\) matrix \(A= (a_{ij})^ n_{i,j= 1}\) with eigenvalues \(\lambda_ 1,\dots, \lambda_ p\), and algebraic and geometric multiplicities \(k_ 1,\dots, k_ p\) and \(g_ 1,\dots, g_ p\), respectively. Let \(h_ i\in N\) be such that \(1\leq h_ i\leq g_ i\), \(i= 1,\dots, p\). Then there exists a matrix \(B= (b_{ij})^ n_{i,j= 1}\) with \(b_{ij}= a_{ij}\), \(j\leq i\) such that \(\lambda_ 1,\dots, \lambda_ p\) are eigenvalues of \(B\) with algebraic and geometric multiplicities \(k_ 1,\dots, k_ p\), and \(h_ 1,\dots, h_ p\), respectively. The paper consists of four sections. A sufficient condition for a set of positive integers \(\{g_ 1, g_ 2,\dots,\) \(g_ n\}\) to be the geometric multiplicities of given eigenvalues for some strictly lower triangular completions of a partial matrix is given. In section 3 the author proposes a method of reducing the investigation of geometric multiplicities of the eigenvalues of a completed matrix to the nilpotent case and proves the main result in this section. Finally, in section 4 the author shows that there are matrices for which Theorem 1 gives the full story, and there are matrices where one may increase some of the geometric multiplicities individually but not simultaneously.