Motzkin–Taussky theorem

The Motzkin–Taussky theorem is a result from operator and matrix theory about the representation of a sum of two bounded, linear operators (resp. matrices). The theorem was proven by Theodore Motzkin and Olga Taussky-Todd.[1]

The theorem is used in perturbation theory, where e.g. operators of the form

are examined.

Statement

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Let   be a finite-dimensional complex vector space. Furthermore, let   be such that all linear combinations

 

are diagonalizable for all  . Then all eigenvalues of   are of the form

 

(i.e. they are linear in   und  ) and   are independent of the choice of  .[2]

Here   stands for an eigenvalue of  .

Comments

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  • Motzkin and Taussky call the above property of the linearity of the eigenvalues in   property L.[3]

Bibliography

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  • Kato, Tosio (1995). Perturbation Theory for Linear Operators. Classics in Mathematics. Vol. 132 (2 ed.). Berlin, Heidelberg: Springer. p. 86. doi:10.1007/978-3-642-66282-9. ISBN 978-3-540-58661-6. 
  • Friedland, Shmuel (1981). "A generalization of the Motzkin-Taussky theorem". Linear Algebra and Its Applications. 36: 103–109. doi:10.1016/0024-3795(81)90223-8. 

Notes

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  1. ^ Motzkin, T. S.; Taussky, Olga (1952). "Pairs of Matrices with Property L". Transactions of the American Mathematical Society. 73 (1): 108–114. doi:10.2307/1990825. JSTOR 1990825. PMC 1063886. PMID 16589359.
  2. ^ Kato, Tosio (1995). Perturbation Theory for Linear Operators. Classics in Mathematics. Vol. 132 (2 ed.). Berlin, Heidelberg: Springer. p. 86. doi:10.1007/978-3-642-66282-9. ISBN 978-3-540-58661-6.
  3. ^ Motzkin, T. S.; Taussky, Olga (1955). "Pairs of Matrices With Property L. II". Transactions of the American Mathematical Society. 80 (2): 387–401. doi:10.2307/1992996. ISSN 0002-9947. JSTOR 1992996.