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

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

are examined.

Statement

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

${\displaystyle T=\alpha A+\beta B}$

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

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

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

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

Comments

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

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.