Editing Ray transfer matrix analysis (section)

== Eigenvalues ==
A ray transfer matrix can be regarded as a [[linear canonical transformation]]. According to the eigenvalues of the optical system, the system can be classified into several classes.{{sfnp|Bastiaans|Alieva|2007}} Assume the ABCD matrix representing a system relates the output ray to the input according to
<math display="block"> \begin{bmatrix}x_2 \\ \theta_2\end{bmatrix} = \begin{bmatrix} A & B \\ C & D \end{bmatrix} \begin{bmatrix}x_1 \\ \theta_1\end{bmatrix}
=\mathbf{T}\mathbf{v} .</math>

We compute the eigenvalues of the matrix <math> \mathbf{T} </math> that satisfy eigenequation
<math display="block"> [\boldsymbol{T}-\lambda I] \mathbf{v} = \begin{bmatrix}
A-\lambda & B \\
C & D-\lambda
\end{bmatrix} \mathbf{v} = 0 ,</math>
by calculating the determinant
<math display="block"> \begin{vmatrix}
A-\lambda & B \\
C & D-\lambda
\end{vmatrix} = \lambda^2 - (A+D) \lambda + 1 = 0 .</math>

Let <math>m = \frac{(A+D)}{2}</math>, and we have eigenvalues <math>\lambda_{1}, \lambda_{2}=m \pm \sqrt{m^{2}-1}</math>.

According to the values of <math>\lambda_{1}</math> and <math>\lambda_{2}</math>, there are several possible cases. For example:

# A pair of real eigenvalues: <math>r</math> and <math>r^{-1}</math>, where <math>r\neq1</math>. This case represents a magnifier <math> \begin{bmatrix} r & 0 \\ 0 & r^{-1} \end{bmatrix} </math>
# <math>\lambda_{1}=\lambda_{2}=1</math> or <math>\lambda_{1}=\lambda_{2}=-1</math>. This case represents unity matrix (or with an additional coordinate reverter) <math> \begin{bmatrix} 1 & 0 \\ 0 & 1\end{bmatrix} </math>.
# <math>\lambda_{1}, \lambda_{2}=\pm1</math>. This case occurs if but not only if the system is either a unity operator, a section of free space, or a lens
# A pair of two unimodular, complex conjugated eigenvalues <math>e^{i\theta}</math> and <math>e^{-i\theta}</math>. This case is similar to a separable [[Fractional Fourier transform|Fractional Fourier Transform]].