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Motzkin–Taussky theorem

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Theorem on linear operators
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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.

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

T + x T 1 {\displaystyle T+xT_{1}}

are examined.

Statement

Let X {\displaystyle X} be a finite-dimensional complex vector space. Furthermore, let A , B ∈ B ( X ) {\displaystyle A,B\in B(X)} be such that all linear combinations

T = α A + β B {\displaystyle T=\alpha A+\beta B}

are diagonalizable for all α , β ∈ C {\displaystyle \alpha ,\beta \in \mathbb {C} } . Then all eigenvalues of T {\displaystyle T} are of the form

λ T = α λ A + β λ B {\displaystyle \lambda _{T}=\alpha \lambda _{A}+\beta \lambda _{B}}

(i.e. they are linear in α {\displaystyle \alpha } und β {\displaystyle \beta } ) and λ A , λ B {\displaystyle \lambda _{A},\lambda _{B}} are independent of the choice of α , β {\displaystyle \alpha ,\beta } .

Here λ A {\displaystyle \lambda _{A}} stands for an eigenvalue of A {\displaystyle A} .

Comments

  • Motzkin and Taussky call the above property of the linearity of the eigenvalues in α , β {\displaystyle \alpha ,\beta } property L.

Bibliography

Notes

  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.
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