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.

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 $\alpha ,\beta$.

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

Comments

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

Bibliography

• 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.