Common eigenvectors and quasicommutativity of sets of simultaneously triangularizable matrices (Q1068162)

Let \(M_ 0=\{A_ 1,A_ 2,...,A_ s\}\) be a set of \(n\times n\) matrices all of whose elements and eigenvalues lie in a field F. Define the \(k^{th}\) order commutators by \(M_ k=\{[A_ i,C^{(k-1)}]=A_ iC^{k-1}-C^{k-1}A_ i:C^{(k-1)}\in M_{k-1}\), \(i=1,2,...,s\}\), \(k=1,2,..\).. A property of \(M_ 0\) is hereditary if, whenever the \(A_ i\) are simultaneously similar to block-triangular matrices (with the same partitioning) the property is inherited by all sets \(\{A_ k^{(1)},...,A_ k^{(s)}\}\) of diagonal blocks. The authors prove: Theorem. Given simultaneously triangularizable \(A_ 1,...,A_ s\) the following are equivalent: \(M_ k=0\) for some positive integer k, the matrices \(A_ i\) are simultaneously similar to conformable block- diagonal matrices, all of whose diagonal blocks have a single eigenvalue, and the existence of a common eigenvector for every distinct s-tuple \(\Lambda_ k=(\lambda_ k^{(1)},...,\lambda_ k^{(s)})\) of corresponding eigenvalues \((\lambda_ k^{(j)}\) an eigenvalue of \(A_ j)\) holds hereditarily. The authors make use of a result of the reviewer and \textit{R. C. Thompson} [Duke Math. J. 35, 441-453 (1968; Zbl 0174.318)].