An eigenvector of any operator \(\hat{M}\) is defined as simply a special particular vector \(|λ_M⟩\) such that the only effect of \(\hat{M}\) acting on \(|λ_M⟩\) is to span the vector by some constant \(λ_M\) (called the eigenvalue of \(\hat{M}\)). Any eigenvector \(|λ_M⟩\) and eigenvalue \(λ_M\) of \(\hat{M}\) is defined as
$$\hat{M}|λ_M⟩=λ_M|λ_M⟩.$$
Thus the eigenvectors and eigenvalues of any observable \(\hat{L}\) is defined as
$$\hat{L}|L⟩=L|L⟩.\tag{9}$$
Let’s take the Hermitian conjugate of both sides of Equation (9) to get
$$⟨L|\hat{L}^†=⟨L|L^*.\tag{10}$$
According to Principle 1, any observable \(\hat{L}\) is Hermitian. Since any Hermitian operator \(\hat{H}\) satisfies the equation \(\hat{H}=\(\hat{H}^†\), it follows that \(\hat{L}=\hat{L}^†\) and we can rewrite Equation (10) as
$$⟨L|\hat{L}=⟨L|L^*.\tag{11}$$
Let’s multiply both sides of Equation (9) by the bra \(⟨L|\) and both sides of equation (11) by the ket \(|L⟩\) to get
$$⟨L|\hat{L}|L⟩=L⟨L|L⟩\tag{12}$$
and
$$⟨L|\hat{L}|L⟩=L^*⟨L|L⟩.\tag{13}$$
Subtracting Equation (12) from Equation (13), we get
$$⟨L|\hat{L}|L⟩-⟨L|\hat{L}|L⟩=0=(L^*-L)⟨L|L⟩.\tag{14}$$
In general, \(⟨L|L⟩≠0\) and is equal to a number \(z\). Therefore it follows that, in general, \(L^*-L=0\) and \(L^*=L\). In order for a number \(z\) to equal its own complex conjugate, it must be real with no imaginary part. Thus, in general, the eigenvalue \(L\) of any observable \(\hat{L}\) is always real. This is a good thing; otherwise Principle 3 wouldn’t make any sense. Recall that principle 3 states that the possible measured values of any quantity (i.e. charge, position, electric field, etc.) are the eigenvalues \(L\) of the corresponding observable \(\hat{L}\). But the measured values obtained from an experiment are always real; in order for Principle 3 to be consistent with this fact, the eigenvalues \(L\) better be real too.