TITLE: How prove this linear algebra $AB=BA$?
QUESTION [8 upvotes]: Suppose $A,B\in M_{n}(\Bbb C)$ satisfies for $\forall a,b\in \Bbb C,aA+bB$ is always diagonalizable.
Show that
$$AB=BA.$$

REPLY [5 votes]: It's a non trivial result. It has indeed be the object of papers: 

[1] T. S. Motzkin and Olga Taussky, Pair of matrices with property $L$, Transactions of the American Mathematical Society
Vol. 80, No. 2 (Nov., 1955), pp. 387-401 
[2] Shmuel Friedland, A generalization of the Motzkin-Taussky theorem,  Linear Algebra and Its Applications - LINEAR ALGEBRA APPL , Volume 36, March 1981, Pages 103–109