On the semigroup of standard symplectic matrices and its applications (Q1887618)

A matrix \(Z\in\mathbb{R}^{2n\times 2n}\) is said to be in the standard symplectic form if \(Z\) enjoys a block LU-decomposition in the sense of \(\left[\begin{smallmatrix} A & 0\\ -H & I\end{smallmatrix}\right] Z= \left[\begin{smallmatrix} I & G\\ 0 & A^T\end{smallmatrix}\right]\), where \(A\) is nonsingular and both \(G\) and \(H\) are symmetric and positive definite in \(\mathbb{R}^{n\times n}\). Such a structure arises naturally in the discrete algebraic Riccati equations. The paper contains two results. First, by means of a parameter representation it is shown that the set of all \(2n\times 2n\) standard symplectic matrices is closed under multiplication and, thus, forms a semigroup. Secondly, block LU-decompositions of powers of \(Z\) can be derived in closed form which, in turn, can be employed recursively to induce an effective structure-preserving alorithm for solving Riccati equations.