# 2 \(\times\) 2 Matrices: Manifolds, Realizations, Applications
## 2 \(\times\) 2 Matrices Covering 3 \(\times\) 3 Matrices
### Real \(\mathsf{SL}(2N,R)\) and Symplectic \(\mathsf{Sp}(2N,R)\) Matrices  
Real \(2N\times 2N\)_symplectic_ matrices \(\mathbf{S}\in\mathsf{Sp}(2N,R)\) are even-dimensional and defined in a manner similar to that defining unitary and orthogonal matrices, but with a distinct skew-symmetric metric [10],  
\[\mathbf{S}\,\widetilde{\mathbf{\Omega}}\,\mathbf{S}^{\top}=\widetilde{\mathbf{ \Omega}},\qquad\widetilde{\mathbf{\Omega}}:=\left( \begin{smallmatrix}\mathbf{0 }&-\mathbf{1}\\ \mathbf{1}&\mathbf{0}\end{smallmatrix} \right),\quad\det\mathbf{S}=1, \tag{8}\]  
where \(\mathbf{0}\) and \(\mathbf{1}\) are \(N\times N\) null and unit submatrices. The Greek root of the peculiar name'symplectic' means interwoven,  imbricate, and is rarely used outside Hamiltonian mechanics and mathematics. In the \(2\times 2\) case (\(N=1\)) all matrices of unit determinant are also symplectic, \(\widetilde{\mathbf{\Omega}}\,\mathbf{S}^{\top}=\widetilde{\mathbf{\Omega}} \left( \begin{smallmatrix}a&c\\ b&d\end{smallmatrix} \right)=\left( \begin{smallmatrix}-b&-d\\ a&c\end{smallmatrix} \right)=\left( \begin{smallmatrix}d&-b\\ c&a\end{smallmatrix} \right)\widetilde{\mathbf{\Omega}}=\mathbf{S}^{-1} \widetilde{\mathbf{\Omega}}\), i.e., \(\mathsf{Sp}(2,R)=\mathsf{SL}(2,R)\). This is an accident among low-dimensional groups. Symplectic matrices are indispensable for Hamiltonian mechanics and geometric optics because they embody linear phase space evolution. Positions \(\mathbf{q}(z)\) and momenta \(\mathbf{p}(z)\) in a system ruled by a Hamiltonian \(h(\mathbf{q},\mathbf{p})\) obey the \(2N\) Hamilton equations, written as a single vector equation with the symplectic metric matrix as:  
\[\frac{\mathrm{d}\mathbf{q}}{\mathrm{d}z}=\frac{\partial h}{\partial\mathbf{p} },\quad\frac{\mathrm{d}\mathbf{p}}{\mathrm{d}z}=-\frac{\partial h}{\partial \mathbf{q}}\quad\Longleftrightarrow\quad\frac{\mathrm{d}}{\mathrm{d}z} \left( \begin{matrix}\mathbf{q}\\ \mathbf{p}\end{matrix} \right)+\widetilde{\mathbf{\Omega}}\left( \begin{matrix} \
abla_{\mathbf{q}}\\ \
abla_{\mathbf{p}}\end{matrix} \right)h(\mathbf{q},\mathbf{p})=\mathbf{0}, \tag{9}\]  
where the evolution parameter \(z\) is time in the case of mechanics or, in optics, distance along the setup axis. Let the phase space \(2N\)-vectors be written as \(\mathbf{w}:=\left( \begin{smallmatrix}\mathbf{q}\\ \mathbf{p}\end{smallmatrix} \right)\) and the phase space gradient as \(\
abla_{\mathbf{w}}:=\left( \begin{smallmatrix}\
abla_{\mathbf{q}}\\ \
abla_{\mathbf{p}}\end{smallmatrix} \right)\). A permissible linear transformation of phase space by a matrix \(\mathbf{S}\) acting as \(\mathbf{w}\mapsto\mathbf{w}^{\prime}=\mathbf{S}\,\mathbf{w}\), is such that it conserves the Hamilton Equation (9). Since the phase space gradient transforms as \(\
abla_{\mathbf{w}}\mapsto\
abla_{\mathbf{w}^{\prime}}^{\prime}=\mathbf{S}^{-1 \,\top}\
abla_{\mathbf{w}}\), this conservation condition is evinced multiplying (9) by \(\mathbf{S}\) from the left and introducing \(\mathbf{1}=\mathbf{S}^{\top}\mathbf{S}^{-1\top}\) between \(\widetilde{\mathbf{\Omega}}\) and \(\
abla_{\mathbf{w}}\) to write:  
\[\frac{\mathrm{d}}{\mathrm{d}z}\mathbf{S}\,\mathbf{w}+\mathbf{S}\,\widetilde{ \mathbf{\Omega}}\,\mathbf{S}^{\top}\mathbf{S}^{-1\top}\
abla_{\mathbf{w}}H( \mathbf{w})=\mathbf{0}\quad\Rightarrow\quad\frac{\mathrm{d}}{\mathrm{d}z} \mathbf{w}^{\prime}+\widetilde{\mathbf{\Omega}}\
abla_{\mathbf{w}^{\prime}}^{ \prime}H(\mathbf{w}^{\prime})=\mathbf{0}, \tag{10}\]  
provided that the matrix \(\mathbf{S}\) satisfies (8), i.e., is symplectic. Transformations of phase space (linear and also nonlinear) that preserve the Hamilton equations of motion are called _canonical_. Basically equivalent definitions of canonicity are provided by transformations that keep invariant the volume and orientation of phase space elements, and those that keep invariant the Poisson brackets between the phase space coordinates [11] [Ch. 3]. All linear canonical transformations are produced through this action of real symplectic matrices.