I have this linear algebra problem in the context of quantum mechanics. Let be a family of linear operators so to each we have a linear operator where is a complex vector space if one is unfamiliar with functional analysis (like I am) or is a Hilbert space if one is. Let’s suppose that this family is differentiable.

Suppose further that is always a Hermitian operator. Suppose that has a discrete spectrum of eigenvalues . I need to show the following:

**Theorem**

Now here is a “proof,” it is not quite rigorous since there are probably a lot of technical details regarding functional analysis that I’m missing out on but:

**Proof** We begin by differentiating the eigenvalue equation with respect to using the product rule:

After multiplying by and rearranging terms we have the following:

Now we can take the adjoint of both sides of the eigenvalue equation to get that since because the eigenvalues of a normal operator are real. This implies that the difference on the right hand side of the above equation is zero. Supposing the eigenvectors normalised, we have . Putting everything together gives the desired result. Now I’m curious, if there are any experts on functional analysis reading this what details have I missed out on?

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