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Thus the matrices must be ordered appropriately, with the last matrix premultiplying the second last, and so on until the first matrix is premultiplied by the second. Other matrices can be constructed to represent interfaces with media of different [[refractive index|refractive indices]], reflection from [[mirror]]s, etc.
== Eigenvalues of Ray Transfer Matrix ==
A ray transfer matrix can been ragarded as [[Linear canonical transformation]]. According to the eigenvalues of the optical system, the system can be classified into several classes<ref>{{Cite journal|last=Bastiaans|first=Martin J.|last2=Alieva|first2=Tatiana|date=2007-03-14|title=Classification of lossless first-order optical systems and the linear canonical transformation|url=http://dx.doi.org/10.1364/josaa.24.001053|journal=Journal of the Optical Society of America A|volume=24|issue=4|pages=1053|doi=10.1364/josaa.24.001053|issn=1084-7529}}</ref>. Assume the the ABCD matrix representing a system relates the output ray to the input according to
<math> \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 eigenvlaues of the matrix <math> \mathbf{T} </math> that satisfy eigenequation
<math> [\boldsymbol{T}-\lambda I] \mathbf{v}=\left[\begin{array}{cc}
A-\lambda & B \\
C & D-\lambda
\end{array}\right]\mathbf{v}=0 </math>,
by calculating the determinant
<math> \left|\begin{array}{cc}
A-\lambda & B \\
C & D-\lambda
\end{array}\right| = \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 Transformer]].
== Table of ray transfer matrices ==
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