# Checking my understanding of 'collapsing into eigenfunctions'?

Hello could someone please check if my understanding of 'collapsing into eigenfunctions' is correct?

Say we have an observable, given by a linear self-adjoint operator $$A$$ and then we have $$\{\psi_n\}_{n\geq 1}$$ as a basis of eigenfunctions, such that $$A\psi_n=c_n\psi_n$$, for $$\mathcal{H}.$$

Suppose I have a normalised wave function in $$\mathcal{H}$$ given by $$\psi(x)=a_1\psi_1+a_2\psi_2+a_3\psi_3.$$ I should say $$a_i's$$ are just the coefficients for these eigenfunctions. For the sake of argument suppose $$c_1=c_3$$, that is, the eigenvalues for $$\psi_1$$ and $$\psi_3$$ are the same. Then after measurement, suppose I obtain $$c_1$$, then am I in the position of claiming the following:

the wavefunction after the measurement is now $$a_1\psi_1+a_3\psi_3.$$ This wave function is no longer normalised unless $$a_2=0.$$

Since it's mathematically convenient to work with normalized vectors, typically one would simply re-normalize $$\psi$$ after such a measurement, but again this is purely a matter of convenience.