Motzkin–Taussky theorem

Template:Technical 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 + x T_{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 = α A + β B

are diagonalizable for all $α, β \in ℂ$ . Then all eigenvalues of $T$ are of the form

λ_{T} = α λ_{A} + β λ_{B}

(i.e. they are linear in $α$ und $β$ ) and $λ_{A}, λ_{B}$ are independent of the choice of $α, β$ .^[2]

Here $λ_{A}$ stands for an eigenvalue of $A$ .

Motzkin and Taussky call the above property of the linearity of the eigenvalues in $α, β$ property L.^[3]