Nonsingular factors of polynomial matrices and (A,B)-invariant subspaces

E. Emre

Research output: Book/ReportReportAcademic

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Abstract

Given a polynomial matrix B(s), we consider the class of nonsingular polynomial matrices L(s) such that B(s) = R(s)L(s) for some polynomial matrix R(s). It is shown that finding such factorizations is equivalent to finding (A,B)-invariant subspaces in the kernel of C where A,B,C are linear maps determined by B(s). In particular, the results yield, as a corollary, a method to determine simultaneously a row proper greatest right divisor of a left invertible polynomial matrix as well as the resulting polynomial matrix whose greatest right divisors are unimodular. The results also relate, the same way, such subspaces of constant systems (C,A,B) where (C,A) is observable and (A,B) is reachable, to the nonsingular right factors of the numerator polynomial matrices in coprime factorizations of the form D^-1 (s)B(s) of their transfer matrices.
Original languageEnglish
Place of PublicationEindhoven
PublisherTechnische Hogeschool Eindhoven
Number of pages14
Publication statusPublished - 1978

Publication series

NameMemorandum COSOR
Volume7812
ISSN (Print)0926-4493

Fingerprint

Polynomial Matrices
Invariant Subspace
Divisor
Coprime Factorization
Invertible matrix
Nonsingular or invertible matrix
Numerator
Linear map
Transfer Matrix
Factorization
Corollary
Subspace
kernel

Cite this

Emre, E. (1978). Nonsingular factors of polynomial matrices and (A,B)-invariant subspaces. (Memorandum COSOR; Vol. 7812). Eindhoven: Technische Hogeschool Eindhoven.
Emre, E. / Nonsingular factors of polynomial matrices and (A,B)-invariant subspaces. Eindhoven : Technische Hogeschool Eindhoven, 1978. 14 p. (Memorandum COSOR).
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abstract = "Given a polynomial matrix B(s), we consider the class of nonsingular polynomial matrices L(s) such that B(s) = R(s)L(s) for some polynomial matrix R(s). It is shown that finding such factorizations is equivalent to finding (A,B)-invariant subspaces in the kernel of C where A,B,C are linear maps determined by B(s). In particular, the results yield, as a corollary, a method to determine simultaneously a row proper greatest right divisor of a left invertible polynomial matrix as well as the resulting polynomial matrix whose greatest right divisors are unimodular. The results also relate, the same way, such subspaces of constant systems (C,A,B) where (C,A) is observable and (A,B) is reachable, to the nonsingular right factors of the numerator polynomial matrices in coprime factorizations of the form D^-1 (s)B(s) of their transfer matrices.",
author = "E. Emre",
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Emre, E 1978, Nonsingular factors of polynomial matrices and (A,B)-invariant subspaces. Memorandum COSOR, vol. 7812, Technische Hogeschool Eindhoven, Eindhoven.

Nonsingular factors of polynomial matrices and (A,B)-invariant subspaces. / Emre, E.

Eindhoven : Technische Hogeschool Eindhoven, 1978. 14 p. (Memorandum COSOR; Vol. 7812).

Research output: Book/ReportReportAcademic

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AB - Given a polynomial matrix B(s), we consider the class of nonsingular polynomial matrices L(s) such that B(s) = R(s)L(s) for some polynomial matrix R(s). It is shown that finding such factorizations is equivalent to finding (A,B)-invariant subspaces in the kernel of C where A,B,C are linear maps determined by B(s). In particular, the results yield, as a corollary, a method to determine simultaneously a row proper greatest right divisor of a left invertible polynomial matrix as well as the resulting polynomial matrix whose greatest right divisors are unimodular. The results also relate, the same way, such subspaces of constant systems (C,A,B) where (C,A) is observable and (A,B) is reachable, to the nonsingular right factors of the numerator polynomial matrices in coprime factorizations of the form D^-1 (s)B(s) of their transfer matrices.

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Emre E. Nonsingular factors of polynomial matrices and (A,B)-invariant subspaces. Eindhoven: Technische Hogeschool Eindhoven, 1978. 14 p. (Memorandum COSOR).