Motzkin–Taussky theorem explained

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

T = \alpha A + \beta B

are diagonalizable for all

\alpha,\beta\in \C . Then all eigenvalues of T are of the formλ _{T} = \alpha λ _{A} + \beta λ _{B}

(i.e. they are linear in

\alpha und \beta ) and λ _{A} , λ _{B} are independent of the choice of \alpha,\beta .^[2]

Here

λ _{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.^[3]

Bibliography

• Book: Kato, Tosio . Tosio Kato . Perturbation Theory for Linear Operators . Classics in Mathematics . Springer . 1995 . 132 . 978-3-540-58661-6 . 2 . Berlin, Heidelberg . 86 . en . 10.1007/978-3-642-66282-9.  
• Friedland . Shmuel . 1981 . A generalization of the Motzkin-Taussky theorem . . 36 . 103–109 . 10.1016/0024-3795(81)90223-8. free .  

Notes and References

1. Motzkin . T. S. . Taussky . Olga . 1952 . Pairs of Matrices with Property L . Transactions of the American Mathematical Society . 73 . 1 . 108–114 . 10.2307/1990825 . 1990825 . 16589359 . 1063886 .
2. Book: Kato, Tosio . Perturbation Theory for Linear Operators . Classics in Mathematics . Springer . 1995 . 132 . 978-3-540-58661-6 . 2 . Berlin, Heidelberg . 86 . en . 10.1007/978-3-642-66282-9.
3. Motzkin . T. S. . Taussky . Olga . 1955 . Pairs of Matrices With Property L. II . Transactions of the American Mathematical Society . 80 . 2 . 387–401 . 10.2307/1992996 . 1992996 . 0002-9947.