The homogeneous projective transformation of general quadratic matrix equations

Abstract

Algebraic quadratic equations are a special case of a single generalized algebraic quadratic matrix equation (GQME). Hence, the importance of that equation in science and engineering is evident. This paper focus on the study of solutions of that GQME and a unified framework for the characterization and identification of solutions at infinity and of finite solutions of general quadratic algebraic matrix equations is presented. The analysis is based on the concept of homogeneous projective transformation for general polynomial systems (Morgan, 1986). In addition, a numerical error analysis for the computed solutions is provided for the assessment of numerical accuracy, stability and conditioning of the computed solutions. The proposed framework is independent of any numerical method and therefore it can be used along with various possible numerical methods for the GQME solution, especially matrix flow-based algorithms (Chu, 1994) (e.g. continuation/homotopy, Morgan, 1989).