Usually, while conducting a measurement on a qubit we are using two projectors, namely $P_0 = |0\rangle \langle 0|$ and $P_1 = |1\rangle \langle 1 |$.
For the case of $P_0$ we have two possible eigenvalues:
- 0, with eigenvector $\begin{pmatrix} 0 \\ 0 \end{pmatrix}$,
- 1, with eigenvector $\begin{pmatrix} 1 \\ 0 \end{pmatrix} = |0\rangle$.
In the case of $P_1$ we have simmilar eigenvalues:
- 0, with eigenvector $\begin{pmatrix} 0 \\ 0 \end{pmatrix}$,
- 1, with eigenvector $\begin{pmatrix} 0 \\ 1 \end{pmatrix} = |1\rangle$.
I guess, that due to the noralization condition we cannot obtain the $\begin{pmatrix} 0 \\ 0 \end{pmatrix}$ states. So the only possible outcomes of a measurement are states $|0\rangle$ and $|1\rangle$ both with eigenvalue 1.
So what is the interpretation of this eigenvalue? Is it something like the "amount of information" stored in a qubit?