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
are examined.
Statement Let be a finite-dimensional complex vector space. Furthermore, let be such that all linear combinations
are diagonalizable for all . Then all eigenvalues of are of the form
(i.e. they are linear in und ) and are independent of the choice of .
Here stands for an eigenvalue of .
Comments Motzkin and Taussky call the above property of the linearity of the eigenvalues in property L.
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Notes