# Kernel Density Estimation on the Siegel Space with an Application to Radar Processing +
## 2 The Siegel Space
### The Siegel Upper Half Space  
The Siegel upper half space is a generalization of the Poincare upper half space (see [23]) for a description of the hyperbolic space. Let \(Sym(n)\) be the space of real symmetric matrices of size \(n\times n\) and \(Sym_{+}(n)\) the set of real symmetric positive definite matrices of size \(n\times n\). The Siegel upper half space is defined by  
\[\mathcal{H}_{n}=\left\{Z=X+iY|X\in Sym(n),Y\in Sym_{+}(n)\right\}.\]  
\(\mathcal{H}_{n}\) is equipped with the following metric:  
\[ds=2 tr(Y^{-1}dZY^{-1}d\overline{Z}).\]  
The set of real symplectic matrices \(Sp(n,\mathbb{R})\) is defined by  
\[g\in Sp(n,\mathbb{R})\Leftrightarrow g^{t}Jg=J,\]  
where  
\[J=\begin{pmatrix}0&I_{n}\\ -I_{n}&0\end{pmatrix},\]  
and \(I_{n}\) is the \(n\times n\) identity matrix. \(Sp(n,\mathbb{R})\) is a subgroup of \(SL_{2n}(\mathbb{R})\), the set of \(2n\times 2n\) invertible matrices of determinant 1. Let \(g=\begin{pmatrix}A&B\\ C&D\end{pmatrix}\in Sp(n,\mathbb{R})\). The metric \(ds\) is invariant under the following action of \(Sp(n,\mathbb{R})\),  
\[g.Z=(AZ+B)(CZ+D)^{-1}.\]  
This action is transitive, i.e.,  
\[\forall Z\in\mathcal{H}_{n},\exists g\in Sp(n,\mathbb{R}),g.i1=Z.\]  
The stabilizer \(K\) of \(iI\) is the set of elements \(g\) of \(Sp(n,\mathbb{R})\) whose action leaves \(i1\) fixed. \(K\) is a subgroup of \(Sp(n,\mathbb{R})\) called the isotropy group. We can verify that  
\[K=\left\{\begin{pmatrix}A&B\\ -B&A\end{pmatrix},A+iB\in SU(n)\right\}.\]A symmetric space is a Riemannian manifold, where the reversal of the geodesics is well defined and is an isometry. Formally, \(exp_{p}(u)\mapsto exp_{p}(-u)\) is an isometry for each \(p\) on the manifold, where \(u\) is a vector in the tangent space at \(p\), and \(exp_{p}\) the Riemannian exponential application at \(p\). In other words, the symmetry around each point is an isometry. \(\mathcal{H}_{n}\) is a symmetric space (see [20]). The structure of a symmetric space can be studied through its isometry group and the Lie algebra of its isometry group. The present work will make use of the Cartan and Iwasawa decompositions of the Lie algebra of \(Sp(n,\mathbb{R})\) (see [22]). Let \(\mathfrak{sp}(n,\mathbb{R})\) be the Lie algebra of \(Sp(n,\mathbb{R})\). Given \(A\), \(B\) and \(C\) three real \(n\times n\) matrices, let denote \(\begin{pmatrix}A&B\\ C&-A^{t}\end{pmatrix}=(A,B,C)\). We have  
\[\mathfrak{sp}(n,\mathbb{R})=\left\{(A,B,C)\left|B\text{ and }C\text{ symmetric}\right.\right\}.\]  
The Cartan decomposition of \(\mathfrak{sp}(n,\mathbb{R})\) is given by  
\[\mathfrak{sp}(n,\mathbb{R})=\mathfrak{t}\oplus\mathfrak{p},\]  
where  
\[\mathfrak{t}=\left\{(A,B,-B)\left|B\text{ symmetric and }A\text{ skew-symmetric}\right.\right\},\]  
\[\mathfrak{p}=\left\{(A,B,B)\left|A,B,\text{ symmetric}\right.\right\}.\]  
The Iwasawa decomposition is given by  
\[\mathfrak{sp}(n,\mathbb{R})=\mathfrak{t}\oplus\mathfrak{a}\oplus\mathfrak{n},\]  
where  
\[\mathfrak{a}=\left\{(A,0,0)\left|H\text{ diagonal}\right.\right\},\]  
\[\mathfrak{n}=\left\{(A,B,0)\left|A\text{ upper triangular with }0\text{ on the diagonal },B\text{ symmetric}\right.\right\}.\]  
It can be shown that  
\[\mathfrak{p}=\cup_{k\in K}Ad_{k}(\mathfrak{a}),\]  
where \(Ad\) is the adjoint representation of \(Sp(n,\mathbb{R})\).