Solving the two-dimensional CIS problem by a rational algorithm

The CIS problem is formulated as follows. Let p be a fixed integer, 1⩽p<n. For given n×n compex matrices A and B, can one verify whether A and B have a common invariant subspace of dimension p by a procedure employing a finite number of arithmetical operations? We describe an algorithm solving th...

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Published inLinear algebra and its applications Vol. 312; no. 1; pp. 115 - 123
Main Authors Al'pin, Yurii A., George, Alan, Ikramov, Khakim D.
Format Journal Article
LanguageEnglish
Published Elsevier Inc 15.06.2000
Subjects
2-generated matrix algebra
Common invariant subspace
Radical
Rational algorithm
Shemesh's theorem
Socle
Common invariant subspace
Rational algorithm
2-generated matrix algebra
Shemesh's theorem
Socle
Radical
Online AccessGet full text
ISSN0024-3795
1873-1856
DOI10.1016/S0024-3795(00)00098-7

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Abstract The CIS problem is formulated as follows. Let p be a fixed integer, 1⩽p<n. For given n×n compex matrices A and B, can one verify whether A and B have a common invariant subspace of dimension p by a procedure employing a finite number of arithmetical operations? We describe an algorithm solving the CIS problem for p=2. Unlike the algorithm proposed earlier by the second and third authors, the new algorithm does not impose any restrictions on A and B. Moreover, when A and B generate a semisimple algebra, the algorithm is able to solve the CIS problem for any p, 1<p<n.
AbstractList The CIS problem is formulated as follows. Let p be a fixed integer, 1⩽p<n. For given n×n compex matrices A and B, can one verify whether A and B have a common invariant subspace of dimension p by a procedure employing a finite number of arithmetical operations? We describe an algorithm solving the CIS problem for p=2. Unlike the algorithm proposed earlier by the second and third authors, the new algorithm does not impose any restrictions on A and B. Moreover, when A and B generate a semisimple algebra, the algorithm is able to solve the CIS problem for any p, 1<p<n.
Author George, Alan
Ikramov, Khakim D.
Al'pin, Yurii A.
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Cites_doi 10.1016/S0024-3795(98)10150-7
10.1016/S0024-3795(99)00252-9
10.1016/0024-3795(84)90085-5
10.1016/0024-3795(78)90043-5
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Issue 1
Keywords Common invariant subspace
Rational algorithm
2-generated matrix algebra
Shemesh's theorem
Socle
Radical
Language English
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References Barker, Eifler, Kezlan (BIB2) 1978; 20
George, Ikramov (BIB3) 1999; 287
Yu.A. Alpin, L. Elsner, Kh.D. Ikramov, On condensed forms for partially commuting matrices, Linear Algebra Appl. 306 (2000) 165–182
A. George, Kh.D. Ikramov, Making the non-commutative spectral theorem a finite criterion, Linear Algebra Appl. (submitted)
Pierce (BIB5) 1982
Shemesh (BIB6) 1984; 62
10.1016/S0024-3795(00)00098-7_BIB1
George (10.1016/S0024-3795(00)00098-7_BIB3) 1999; 287
Shemesh (10.1016/S0024-3795(00)00098-7_BIB6) 1984; 62
10.1016/S0024-3795(00)00098-7_BIB4
Barker (10.1016/S0024-3795(00)00098-7_BIB2) 1978; 20
Pierce (10.1016/S0024-3795(00)00098-7_BIB5) 1982
References_xml – reference: Yu.A. Alpin, L. Elsner, Kh.D. Ikramov, On condensed forms for partially commuting matrices, Linear Algebra Appl. 306 (2000) 165–182
– reference: A. George, Kh.D. Ikramov, Making the non-commutative spectral theorem a finite criterion, Linear Algebra Appl. (submitted)
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  publication-title: Linear Algebra Appl.
  doi: 10.1016/0024-3795(78)90043-5
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SubjectTerms 2-generated matrix algebra
Common invariant subspace
Radical
Rational algorithm
Shemesh's theorem
Socle
Title Solving the two-dimensional CIS problem by a rational algorithm
URI https://dx.doi.org/10.1016/S0024-3795(00)00098-7
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