Wavelet and Linear Algebra

Abstract This paper explores the Lyapunov, quadratic matrix, and matrix square root equations under normality assumptions on the system matrix, utilizing the Moore-Penrose inverse ($A^\dagger$) for solvability in singular cases. We provide a detailed historical overview and extend our spectral decomposition method to achieve unique solutions for these equations, building on prior work for invertible matrices. The proposed approach handles singular normal matrices via orthogonality conditions on the kernel, and numerical examples confirm its accuracy and efficiency compared to conventional methods.