Skip to main content Contents Index Prev Up Next \(\newcommand{\vf}[1]{\mathbf{\boldsymbol{\vec{#1}}}}
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Section 4.4 Normalization of Eigenvectors
In
Section 3.7 , we defined the inner product operation on abstract vectors with complex components, such as
\begin{equation}
\vert v\rangle
\doteq\begin{pmatrix}a\\b\\ \vdots
\end{pmatrix} \text{.}\tag{4.4.1}
\end{equation}
If we take the inner product of this vector with itself,
\begin{align}
\langle v\vert v\rangle
\amp = \begin{pmatrix}a^* \amp b^* \amp \dots
\end{pmatrix} \begin{pmatrix}a\\b\\ \vdots
\end{pmatrix}\notag\\
\amp = \vert a\vert^2 + \vert b\vert^2 +\dots\text{,}\tag{4.4.2}
\end{align}
the operation always yields a positive, real number. Thus, we can use the square root of this operation \(\vert \vert v\rangle\vert\text{.}\)
\begin{equation}
\left\vert \vert v\rangle\right\vert
=\left\{\langle v\vert v\rangle\right\}^{\frac{1}{2}}\tag{4.4.3}
\end{equation}
to define the
norm (also called
magnitude or
length ) of the vector. This definition is a natural generalization of the dot product, see
Section 1.7 , of a real vector with itself.
Definition 4.4 . Normalized Vector.
From the eigenvalue/eigenvector equation
(4.1.1) :
\begin{equation}
A \left|v\right> = \lambda \left|v\right>\tag{4.4.4}
\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
\(N\) can pull through to the left on both sides of the equation.
If a choice of \(N\) normalizes a particular vector, then so does any arbitrary complex phase times the original number \(Ne^{i\alpha}\) (for \(\alpha\) real). Usually, we choose \(N\) to be real, for simplicity.
It is always possible to choose the number
\(N\) to rescale the eigenvector to have length
\(1\text{.}\) Such an eigenvector is called
normalized .
