Vali-e-Asr university of Rafsanjan Wavelet and Linear Algebra 2383-1936 13 1 2026 07 01 Generalized Equation Characterizations of Normal Matrices and Applications to Lyapunov, Quadratic, and Matrix Square Root Problems 66 75 737856 10.22072/wala.2026.2075707.1476 EN Alimohammad Nazari Arak university of Iran 0000-0002-3231-0340 Journal Article 2025 10 25 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. https://wala.vru.ac.ir/article_737856_9ea953c00b8109b48edf03408004e148.pdf