Section 4.4 Normalization of Eigenvectors
From the eigenvalue/eigenvector equation:
\begin{equation}
A \left|v\right> = \lambda \left|v\right>\tag{4.4.1}
\end{equation}
it is straightforward to show that if \(\vert v\rangle\) is an eigenvector of \(A\text{,}\) then, any multiple \(N\vert v\rangle\) of \(\vert v\rangle\) is also an eigenvector since the (real or complex) number \(N\) can pull through to the left on both sides of the equation.
Notice that for any vector \(\vert v\rangle=\begin{pmatrix}a\\b \end{pmatrix}\text{,}\) the operation:
\begin{align}
\langle v\vert v\rangle
\amp = \begin{pmatrix}a^* \amp b^*
\end{pmatrix} \begin{pmatrix}a\\b
\end{pmatrix}\notag\\
\amp = \vert a\vert^2 + \vert b\vert^2\tag{4.4.2}
\end{align}
always yields a positive, real number. Thus, we can use the square root of this operation to define the norm or length of the vector, \(\vert \vert v\rangle\vert\text{.}\)
\begin{equation}
\vert \vert v\rangle\vert=\left\{\langle v\vert v\rangle\right\}^{\frac{1}{2}}\tag{4.4.3}
\end{equation}
It is always possible to choose the number \(N\) above to find an eigenvector with length \(1\text{.}\) Such an eigenvector is called normalized.