In Mathematics and Logic: Retrospect and Prospects (Kac & Ulam, 1969) I came across the following curious claim on p. 79:
One can show that in an $n$-dimensional vector space there are $n$ linearly independent vectors $E_1, E_2, \dots , E_n$ of this kind. They are called eigenvectors of the transformation $T$ and the corresponding scalars $\lambda_1, \lambda_2, \dots , \lambda_n$ are called the eigenvalues.
The claim seems to be that every matrix can be diagonalized. (More context doesn't help.) This is clearly false -- so what is going on here? Am I misunderstanding the claim, or is this just an error?