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]

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## Abstract

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.

Item Type: | MIMS Preprint |
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Uncontrolled Keywords: | eigenvalues, index sum theorem, invariant orders, minimal indices, poles, polynomial matrices, rational matrices, structural indices, zeros |

Subjects: | MSC 2010, the AMS's Mathematics Subject Classification > 15 Linear and multilinear algebra; matrix theory MSC 2010, the AMS's Mathematics Subject Classification > 65 Numerical analysis MSC 2010, the AMS's Mathematics Subject Classification > 93 Systems theory; control |

Depositing User: | Dr. D. Steven Mackey |

Date Deposited: | 18 Feb 2018 08:35 |

Last Modified: | 18 Feb 2018 08:35 |

URI: | https://eprints.maths.manchester.ac.uk/id/eprint/2624 |

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