Matrices with prescribed rows and invariant factors
This paper contains a most interesting and beautiful result. About nine years ago \textit{E. M. de Sá} [ibid. 24, 33-50 (1979; Zbl 0395.15009)] and \textit{R. C. Thompson} [ibid. 24, 1-31 (1979; Zbl 0395.15003)] found the complete relations between the invariant polynomials of a matrix (over a field) and the invariant polynomials of one of its principal submatrices. Their result is known as Sá-Thompson interlacing inequalities and raised some questions so far unsettled. The present author tackles and solves another problem of the same type: consider the matrix (over a field) \[ G=\left[ \begin{matrix} A\quad B\\ C\quad D\end{matrix} \right] \] where A and D are square blocks, A and B are known and C and D are unknown. Find a necessary and sufficient condition for the existence of C and D so that G has prescribed invariant polynomials. It cannot be said that the condition presented in this paper is too involved to be stated here. What prevents the reviewer from stating it is the need of explaining the notation. However the reviewer would like to mention that the necessary and sufficient condition consists of some interlacing inequalities similar to those of Sá and Thompson (as expected) together with a majorization relation involving the controllability indices of the pair (A,B) and the degrees of certain polynomials.
scientific article; zbMATH DE number 4027674