On the Segre characteristic of a block triangular matrix
Consider two complex matrices \(A\) and \(B\) of sizes \(n\times n\) and \(m\times m\) respectively, and let \(\lambda\) be an eigenvalue of both matrices. If \(\alpha= (n_1,n_2)\) and \(\beta= (m_1,m_2)\) are the Segre characteristics associated with \(\lambda\) of \(A\) and \(B\), respectively, and \(\gamma= (r_1,r_2, r_3,r_4)\) is a nonincreasing sequence of nonnegative integers, then a method for determining when \(\gamma\) is the Segre characteristic of the block triangular matrix \[ M=\left[ \begin{matrix} A & L \\ 0 & B\end{matrix} \right] \] in terms of \(\alpha\) and \(\beta\), and the structure of the matrix \(L\) is presented. A complete explicit description of the Segre characteristic of \(M\) associated with \(\lambda\) is obtained. Using similar techniques, general cases when \(\alpha,\beta\) and \(\gamma\) have more elements and satisfy some size restrictions are considered.
scientific article; zbMATH DE number 1419869