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

T+xT_{1}

are examined.

Statement

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

T=\alpha A+\beta B

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

\lambda _{T}=\alpha \lambda _{A}+\beta \lambda _{B}

(i.e. they are linear in $\alpha$ und $\beta$ ) and $\lambda _{A},\lambda _{B}$ are independent of the choice of $\alpha ,\beta$ .[2]

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

Motzkin and Taussky call the above property of the linearity of the eigenvalues in $\alpha ,\beta$ property L.[3]

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

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.
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.
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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