References: edX online course MIT 8.05.1x Week 4.
Sheldon Axler (2015), Linear Algebra Done Right, 3rd edition, Springer. Chapter 7.
A hermitian operator satisfies . [Axler (and most mathematicians, probably) refers to a hermitian operator as self-adjoint and uses the notation for .]
As preparation for discussing hermitian operators, we need the following theorem.
Theorem 1 If is a linear operator in a complex vector space , then if for all , then .
Proof: The idea is to show something even more general, namely that for all . If we can do this, then setting means that for all , which in turn implies that for all , implying further that .
Zwiebach goes through a few stages in developing the proof, but the end result is that we can write
Note that all the terms on the RHS are of the form for some . Thus if we require for all , then all four terms are separately 0, meaning that as desired, completing the proof.
Although we’ve used the imaginary number in this proof, we might wonder if it really does restrict the result to complex vector spaces. That is, is there some other decomposition of that doesn’t required complex numbers that would still work?
In fact, we don’t need to worry about this, since there is a simple counter-example to the theorem if we consider a real vector space. In 2-d or 3-d space, an operator that rotates a vector through always produces a vector orthogonal to the original, resulting in for all . In this case, so the theorem is definitely not true for real vector spaces.
Now we can turn to a few theorems about hermitian operators. First, since every operator on a finite-dimensional complex vector space has at least one eigenvalue, we know that every hermitian operator has at least one eigenvalue. This leads to the first theorem on hermitian operators.
Theorem 2 All eigenvalues of hermitian operators are real.
Proof: Since at least one eigenvalue exists, let be the corresponding non-zero eigenvector, so that . We have
Since we also have
Equating the last two equations, and remembering that , we have , so is real.
Next, a theorem on the eigenvectors of distinct eigenvalues.
Theorem 3 Eigenvectors associated with different eigenvalues of a hermitian operator are orthogonal.
Proof: Suppose are two eigenvalues of , and and are the corresponding eigenvectors. Then and . Taking an inner product, we have
where in the last line we used the fact that is real when taking it outside the inner product. Equating the first and last lines and using , we see that as required.