Jordan structures of strictly lower triangular completions of nilpotent matrices
The authors prove the following theorem conjectured by \textit{L. Rodman} and \textit{T. Shalom} [Linear Algebra Appl. 168, 221-249 (1992; Zbl 0748.15011)]: Let \(A\) be a nilpotent \(n \times n\) matrix with Jordan blocks of size \(q_1 \geq \cdots \geq q_s\). If \(p_1 \geq \cdots \geq p_r\) is a sequence of positive integers that majorizes the sequence of \(q\)'s (i.e. \(r \leq s\), \(p_1 + p_2 + \cdots + p_t \geq q_1 + q_2 + \cdots + q_t\) for \(t = 1, \dots, r\), and \(p_1 + \cdots + p_r = q_1 + \cdots+q_s)\), then there exists a strictly lower triangular matrix \(T\) such that \(A + T\) is nilpotent and the \(p\)'s are the sizes of the Jordan blocks of \(A + T\).
scientific article; zbMATH DE number 838571
scientific article