On the Necessary and Sufficient Condition for a Set of Matrices to Commute and Some Further Linked Results
This paper investigates the necessary and sufficient condition for a set of (real or complex) matrices to commute. It is proved that the commutator [A,B]=0 for two matrices A and B if and only if a vector v(B) defined uniquely from the matrix B is in the null space of a well-structured matrix defin...
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| Format: | Article |
| Language: | English |
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Hindawi Limited
2009-01-01
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| Series: | Mathematical Problems in Engineering |
| Online Access: | http://dx.doi.org/10.1155/2009/650970 |
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doaj-2a991195d48d46fba0dae85fd2df09112020-11-25T01:13:09ZengHindawi LimitedMathematical Problems in Engineering1024-123X1563-51472009-01-01200910.1155/2009/650970650970On the Necessary and Sufficient Condition for a Set of Matrices to Commute and Some Further Linked ResultsM. De La Sen0Department of Electricity and Electronics, Faculty of Science and Technology, University of the Basque Country, Campus of Leioa (Bizkaia). Aptdo. 644, 48080 Bilbao, SpainThis paper investigates the necessary and sufficient condition for a set of (real or complex) matrices to commute. It is proved that the commutator [A,B]=0 for two matrices A and B if and only if a vector v(B) defined uniquely from the matrix B is in the null space of a well-structured matrix defined as the Kronecker sum A⊕(−A∗), which is always rank defective. This result is extendable directly to any countable set of commuting matrices. Complementary results are derived concerning the commutators of certain matrices with functions of matrices f(A) which extend the well-known sufficiency-type commuting result [A,f(A)]=0.http://dx.doi.org/10.1155/2009/650970 |
| collection |
DOAJ |
| language |
English |
| format |
Article |
| sources |
DOAJ |
| author |
M. De La Sen |
| spellingShingle |
M. De La Sen On the Necessary and Sufficient Condition for a Set of Matrices to Commute and Some Further Linked Results Mathematical Problems in Engineering |
| author_facet |
M. De La Sen |
| author_sort |
M. De La Sen |
| title |
On the Necessary and Sufficient Condition for a Set of Matrices to Commute and
Some Further Linked Results |
| title_short |
On the Necessary and Sufficient Condition for a Set of Matrices to Commute and
Some Further Linked Results |
| title_full |
On the Necessary and Sufficient Condition for a Set of Matrices to Commute and
Some Further Linked Results |
| title_fullStr |
On the Necessary and Sufficient Condition for a Set of Matrices to Commute and
Some Further Linked Results |
| title_full_unstemmed |
On the Necessary and Sufficient Condition for a Set of Matrices to Commute and
Some Further Linked Results |
| title_sort |
on the necessary and sufficient condition for a set of matrices to commute and
some further linked results |
| publisher |
Hindawi Limited |
| series |
Mathematical Problems in Engineering |
| issn |
1024-123X 1563-5147 |
| publishDate |
2009-01-01 |
| description |
This paper investigates the necessary and sufficient condition for a set of (real or complex) matrices to commute. It is proved that the commutator [A,B]=0 for two matrices A and B if and only if a vector
v(B) defined uniquely from the matrix B is in the null space of a well-structured matrix defined as the Kronecker sum A⊕(−A∗), which is always rank defective. This result is extendable directly to any countable set of commuting matrices. Complementary results are derived concerning the commutators of certain matrices with functions of matrices f(A) which extend the well-known sufficiency-type commuting result [A,f(A)]=0. |
| url |
http://dx.doi.org/10.1155/2009/650970 |
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AT mdelasen onthenecessaryandsufficientconditionforasetofmatricestocommuteandsomefurtherlinkedresults |
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