[electricslpnsld]

> The identity applies to “Hermitian” matrices, which transform eigenvectors by real amounts (as opposed to those that involve imaginary numbers), and which thus apply in real-world situations.

This is a weird characterization of Hermitian matrices — this implies that rotations are not ‘real-world’.

[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.

[angry_octet]

It's really bad that the author propagates the idea that complex numbers are not real physical phenomena because maths uses the term 'imaginary'.

[ivalm]

It's a bit different. To have real eigenvalues (ie all observables) you need to have hermetian operators. They were working on the hamiltonian operator, which is hermetian since its eigenvalues are energies (ie are observable and are thus real numbers)

[Koshkin]

Read: real-world situations [as described by Quantum Mechanics]