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

: $\lambda_=\alpha\lambda_ + \beta \lambda_$

(i.e. they are linear in $\alpha$ und $\beta$) and $\lambda_,\lambda_$ are independent of the choice of $\alpha,\beta$.

Here $\lambda_$ 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''.

Bibliography

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Notes

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Mathematical theorems
Linear algebra
Perturbation theory
Linear operators