Van Dooren's Index Sum Theorem and Rational Matrices with Prescribed Structural Data
| Autor: | Froilán M. Dopico, D. Steven Mackey, Richard Hollister, Luis Miguel Anguas |
|---|---|
| Přispěvatelé: | Ministerio de Economía y Competitividad (España) |
| Rok vydání: | 2019 |
| Předmět: |
Left and right
Minimal indices Pure mathematics Invariant orders Matemáticas Pole–zero plot Eigenvalues 010103 numerical & computational mathematics Lambda 01 natural sciences Rational matrices Zeros Index sum theorem Polynomial matrices Computer Science::Symbolic Computation Structural indices Poles 0101 mathematics Analysis Eigenvalues and eigenvectors Mathematics |
| Zdroj: | e-Archivo. Repositorio Institucional de la Universidad Carlos III de Madrid instname e-Archivo: Repositorio Institucional de la Universidad Carlos III de Madrid Universidad Carlos III de Madrid (UC3M) |
| ISSN: | 1095-7162 0895-4798 |
| DOI: | 10.1137/18m1171370 |
| Popis: | The structural data of any rational matrix R(\lambda ), i.e., the structural indices of its poles and zeros together with the minimal indices of its left and right nonespaces, 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 rediscovered) 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 \scrL of prescribed structural data does there exist some rational matrix R(\lambda ) that realizes exactly the list \scrL ? We show that Van Dooren's condition is the only constraint on rational realizability; that is, a list \scrL is the structural data of some rational matrix R(\lambda ) if and only if \scrL satisfies the rational index sum condition. |
| Databáze: | OpenAIRE |
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