# Van Dooren's Index Sum Theorem and Rational Matrices with Prescribed Structural Data

Anguas, Luis M. and Dopico, Froilán M. and Hollister, Richard and Mackey, D. Steven (2018) Van Dooren's Index Sum Theorem and Rational Matrices with Prescribed Structural Data. [MIMS Preprint]

The structural data of any rational matrix $R(\la)$, i.e., the structural indices of its poles and zeros together with the minimal indices of its left and right nullspaces, is known to satisfy a simple condition involving certain sums of these indices. This fundamental constraint was first proved by Van Dooren in $1978$; here we refer to this result as the rational index sum theorem''. An analogous result for polynomial matrices has been independently discovered (and re-discovered) several times in the past three decades. In this paper we clarify the connection between these two seemingly different index sum theorems, describe a little bit of the history of their development, and discuss their curious apparent unawareness of each other. Finally, we use the connection between these results to solve a fundamental inverse problem for rational matrices --- for which lists ${\cal L}$ of prescribed structural data does there exist some rational matrix $R(\la)$ that realizes exactly the list ${\cal L}$? We show that Van Dooren's condition is the \emph{only} constraint on rational realizability; that is, a list ${\cal L}$ is the structural data of some rational $R(\la)$ if and only if ${\cal L}$ satisfies the rational index sum condition.