# Mathematics Colloquia and Seminars

The eigenvalue problem of the Hamiltonian matrices has many applications in practice. The major application to the linear control system and information theory is the solutions of the algebraic Riccati equations. The main challenge of this problem in numerical computation aspect is how to use the algebraic structures of the matrices. In this talk we first show a new backward stable structure preserving numerical algorithm for computing the eigenvalues of the Hamiltonian matrices. The main trick is to compute a symplectic URV like decomposition of a Hamiltonian matrix $cH$. Using this decomposition and the relation between $cH$ and the extended matrix $mat{cc}0&cH\cH&0 ix$, we then develop a numerical method to compute the Lagrangian invariant subspace as well as the solutions of the associated algebraic Riccati equation.