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 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.
Notes