Explicit Jordan form for certain block triangular matrices. II (Q1183210)

This paper is a contribution in the direction of explicitly determining the Jordan canonical form of \(M={B X \choose 0 C}\) in terms of \(X\), given the Jordan structures of the square matrices \(B\) and \(C\). Without loss of generality, \(M\) is assumed to have only 0 as an eigenvalue and \(B\), \(C\) are in Jordan canonical form. In part I [ibid. 150, 297-314 (1991; Zbl 0725.15009)] the authors solved the problem when 0 has geometric multiplicity 1 in \(C\). The present paper tackles the general problem. For each positive integer \(k\), if \(M= ({B^ k \atop 0} {X^{(k)} \atop C^ k})\), the authors define the ``action matrix'' \(A_ k\), which is a block Toeplitz matrix based on diagonal sums of blocks from \(X^{(k)}\) (the blocking is induced by the Jordan blocks in \(B\) and \(C\)) and they prove that \(\text{rank} M^ k=\text{rank} B^ k+\text{rank} C^ k+\text{rank} A_ k\). They show that the number of Jordan blocks in \(M\) is \(\min\{p,q\}+\text{def} A_ 1\), where \(p\) (\(q\)) is the number of Jordan blocks in \(B\) (respectively \(C\)) and \(\text{def} A_ 1\) is the rank deficiency of \(A_ 1\). They also give a formula for the size of the largest Jordan block of \(M\). If all the blocks of \(A_ 1\) have full rank they find an explicit Jordan form for \(M\). They also solve the problem if \(A_ 1\) has full rank and they show that if \(p=q=2\) the problem has a neat solution.