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 $$\alpha,\beta\in \C$$. 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.