# A factorization for $Z \times Z$-matrices yielding solutions of Toda-type hierarchies

G.F. Helminck

Research output: Book/ReportReportProfessional

### Abstract

In this paper one considers the problem of finding solutions to a number of Toda-type hierarchies. All of them are associated with a commutative subalgebra of the $k \times k$-matrices. The first one is formulated in terms of upper triangular $\mathbb{Z} \times \mathbb{Z}$-matrices, the second one in terms of lower triangular ones and the third is a combination of the two foregoing types. It is shown that in an appropriate group setting solutions of the linearization of these Lax equations can be constructed by using the Birkhoff decomposition in the relevant group.
Original language Undefined Enschede University of Twente, Department of Applied Mathematics 16 Published - 2004

### Publication series

Name Memorandum / Department of Applied Mathematics University of Twente, Department of Applied Mathematics 1745 0169-2690

• MSC-20G15
• MSC-20G20
• MSC-22E15
• METIS-227111
• EWI-3565
• IR-65929
• MSC-22E46

### Cite this

Helminck, G. F. (2004). A factorization for $Z \times Z$-matrices yielding solutions of Toda-type hierarchies. (Memorandum / Department of Applied Mathematics; No. 1745). Enschede: University of Twente, Department of Applied Mathematics.
Helminck, G.F. / A factorization for $Z \times Z$-matrices yielding solutions of Toda-type hierarchies. Enschede : University of Twente, Department of Applied Mathematics, 2004. 16 p. (Memorandum / Department of Applied Mathematics; 1745).
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abstract = "In this paper one considers the problem of finding solutions to a number of Toda-type hierarchies. All of them are associated with a commutative subalgebra of the $k \times k$-matrices. The first one is formulated in terms of upper triangular $\mathbb{Z} \times \mathbb{Z}$-matrices, the second one in terms of lower triangular ones and the third is a combination of the two foregoing types. It is shown that in an appropriate group setting solutions of the linearization of these Lax equations can be constructed by using the Birkhoff decomposition in the relevant group.",
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author = "G.F. Helminck",
note = "Imported from MEMORANDA",
year = "2004",
language = "Undefined",
series = "Memorandum / Department of Applied Mathematics",
publisher = "University of Twente, Department of Applied Mathematics",
number = "1745",

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Helminck, GF 2004, A factorization for $Z \times Z$-matrices yielding solutions of Toda-type hierarchies. Memorandum / Department of Applied Mathematics, no. 1745, University of Twente, Department of Applied Mathematics, Enschede.
Enschede : University of Twente, Department of Applied Mathematics, 2004. 16 p. (Memorandum / Department of Applied Mathematics; No. 1745).

Research output: Book/ReportReportProfessional

TY - BOOK

T1 - A factorization for $Z \times Z$-matrices yielding solutions of Toda-type hierarchies

AU - Helminck, G.F.

N1 - Imported from MEMORANDA

PY - 2004

Y1 - 2004

N2 - In this paper one considers the problem of finding solutions to a number of Toda-type hierarchies. All of them are associated with a commutative subalgebra of the $k \times k$-matrices. The first one is formulated in terms of upper triangular $\mathbb{Z} \times \mathbb{Z}$-matrices, the second one in terms of lower triangular ones and the third is a combination of the two foregoing types. It is shown that in an appropriate group setting solutions of the linearization of these Lax equations can be constructed by using the Birkhoff decomposition in the relevant group.

AB - In this paper one considers the problem of finding solutions to a number of Toda-type hierarchies. All of them are associated with a commutative subalgebra of the $k \times k$-matrices. The first one is formulated in terms of upper triangular $\mathbb{Z} \times \mathbb{Z}$-matrices, the second one in terms of lower triangular ones and the third is a combination of the two foregoing types. It is shown that in an appropriate group setting solutions of the linearization of these Lax equations can be constructed by using the Birkhoff decomposition in the relevant group.

KW - MSC-20G15

KW - MSC-20G20

KW - MSC-22E15

KW - METIS-227111

KW - EWI-3565

KW - IR-65929

KW - MSC-22E46

M3 - Report

T3 - Memorandum / Department of Applied Mathematics

BT - A factorization for $Z \times Z$-matrices yielding solutions of Toda-type hierarchies

PB - University of Twente, Department of Applied Mathematics

CY - Enschede

ER -

Helminck GF. A factorization for $Z \times Z$-matrices yielding solutions of Toda-type hierarchies. Enschede: University of Twente, Department of Applied Mathematics, 2004. 16 p. (Memorandum / Department of Applied Mathematics; 1745).