The contragredient equivalence for several matrices: A set of invariants
An \(n\)-tuple of complex matrices \((A_1,A_2,\dots,A_n)\) is said to be cyclically multiplicable if the products \(A_iA_{i+1}\) are defined for all \(i\) modulo \(n\). Two cyclically multiplicable \(n\)-tuples of matrices \((A_1,A_2,\dots,A_n)\) and \((A_1',A_2',\dots,A_n')\) are contragrediently equivalent if there exist nonsingular matrices \(T_1,T_2,\dots,T_n\) such that \(A_i'=T_i^{-1}A_iT_{i+1}\) for all \(i\) modulo \(n\). When this holds, clearly the square matrices \(B_i=A_iA_{i+1}\dots A_{i-1}\) and \(B_i'=A_i'A_{i+1}'\dots A_{i-1}'\) are similar for all \(i\) modulo \(n\). The main problem studied and solved in a natural but quite technical manner in this well written paper is to find a necessary and sufficient condition for two cyclically multiplicable \(n\)-tuples to be contragrediently equivalent. The condition is that they have the same ``spectra of divisors''. Unfortunately a few pages of technical definitions and preliminary results are needed to fully explain the result. This work builds on substantial earlier work by several authors (including the present one), beginning with the fundamental paper by \textit{H. Flanders}, [Proc. Am. Math. Soc. 2, 871-874 (1951; Zbl 0044.00602)].
scientific article; zbMATH DE number 1331314