Motzkin–Taussky theorem

Short description: Theorem on linear operators

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

[math]\displaystyle{ T+xT_1 }[/math]

are examined.

Statement

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

[math]\displaystyle{ T=\alpha A+\beta B }[/math]

are diagonalizable for all [math]\displaystyle{ \alpha,\beta\in \C }[/math]. Then all eigenvalues of [math]\displaystyle{ T }[/math] are of the form

[math]\displaystyle{ \lambda_{T}=\alpha\lambda_{A} + \beta \lambda_{B} }[/math]

(i.e. they are linear in [math]\displaystyle{ \alpha }[/math] und [math]\displaystyle{ \beta }[/math]) and [math]\displaystyle{ \lambda_{A},\lambda_{B} }[/math] are independent of the choice of [math]\displaystyle{ \alpha,\beta }[/math].

Here [math]\displaystyle{ \lambda_{A} }[/math] stands for an eigenvalue of [math]\displaystyle{ A }[/math].

Comments

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

Bibliography

• 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.