2. Theoretical Analysis

This chapter provides a survey of the main theoretical properties concerning algebraic Riccati equations and related matrix equations, together with the description of the main tools for the design and analysis of solution algorithms. We start with the description of the interplay between graph deflating subspaces of certain matrix pencils and algebraic Riccati equations, then we deal with spectral properties of the solutions of algebraic Riccati equations with specific attention addressed to extremal and critical solutions. The role of the shift techniques for overcoming the computational problems encountered in the critical cases is addressed. The chapter continues with the description and analysis of transformations of matrix equations. We deal with transformations which map a CARE to a DARE and, conversely, and transformations which map NAREs to UQMEs. The chapter is concluded by reporting some perturbation results for the matrix equations studied in the book.