Why are self-adjoint operators important?

Question

I am learning about self-adjoint and normal operators.

So far, they have come up in the Spectral theorem, which says self-adjoint operators have an eigenvalue basis and a corresponding diagonal matrix.

Do self-adjoint (or normal) operators have any other useful properties? (i.e. why are they important in linear algebra?)

Since it is fairly straightforward, at least from what i've learnt, to find eigenvalues with the characteristic equation, it seems to me that you wouldn’t first try to see if an operator is self-adjoint before you diagonalize it, but would just find the eigenvalues and vectors.

Answer 1

Addressing your second question first, finding exact eigenvalues becomes quite difficult and computationally intensive as the size of the matrices involved increases. In particular, there is (provably!) no general formula for finding the roots of a polynomial of degree 5 or higher, so we might not even be able to explicitly find the eigenvalues for a matrix $5 \times 5$ or larger! Finding efficient numerical algorithms for approximating eigenvalues of large matrices is a huge field---for example, the original implementation of Google's search algorithm is built around approximating the dominant eigenvalue and eigenvector for the adjacency matrix of the entire internet.

As for why someone might care about self-adjoint operators, they take a pivotal role in physics and especially quantum mechanics. Given a physical system, all of the observables that we might ask (position, momentum, etc) are represented by self-adjoint operators in some appropriate context. For example, Heisenberg's uncertainty principle is often stated as $$PQ - QP = i\hbar$$ where P and Q are the (self-adjoint!) position and momentum operators.