[cshimmin]

Rotation matrices are hermitian. It's a popular science article, but the author is alluding to the fact that Hermitian matrices have real eignenvalues. In quantum mechanics and quantum field theory, all _physically observable quantities_ of a system can be expressed as the eigenvalues of a set of a (matrix) operator associated with that quantity acting on basis states in the Hilbert space of the system. This makes sense a posteriori since we have no idea how we would interpret e.g. complex-valued energy or momentum.

Edit: electricslpnsld correctly points out that the first sentence is false, see below.

[electricslpnsld]

> Rotation matrices are hermitian

Is that the case? Consider

|a -b|

|b a|

with Eigenvalues lambda = a +/- i b

a = cos theta and b = sin theta gives a rotation, so the Eigenvalues are complex.

[contravariant]

Yeah rotation matrix are not hermitian (note that you don't need to show the eigenvalues aren't real, you just need to show it's not self-conjugate). The OP may have been confused by the fact that you can make a rotation matrix of eigenvectors of a hermitian matrix, which diagonalises the original matrix into two conjugate rotation matrices with a diagonal matrix between them.

[cshimmin]

D'oh you are correct! It's a bit late here. Anyways the rest of my point stands about the author's intent. To connect that to your original comment: a rotation is not "observable", instead, it's a transformation that modifies a physical system. In quantum mechanics, physical observables (energy, momentum, position, mass, etc) are always associated with Hermitian operators that act on the Hilbert space of states for the system.