A Numerical Investigation of Spectral Stability and Eigenvalue Distribution in Non-Symmetric Random Matrices: A Pseudospectra Approach Using MATLAB

Abstract

Random Matrix Theory (RMT) serves as a fundamental mathematical tool for modeling and understanding the behavior of complex systems in physics and engineering. This paper presents an in-depth numerical and analytical investigation into the eigenvalue distribution and spectral stability of non-symmetric random matrices, focusing specifically on verifying the Circular Law and the phenomenon of non-normality. The research methodology involved developing a high-precision computational framework using the MATLAB simulation environment; large-scale simulations were conducted on matrices of varying dimensions (N = 100, 400, 1000) to track spectral behavior from finite-size effects through to the large-limit regime. To overcome the limitations of traditional spectral analysis in detecting sensitivity to perturbations, the study employs pseudospectra as an advanced diagnostic tool to identify regions of algebraic instability in the complex plane. The results demonstrate close agreement between the empirical spectral distribution and the theoretical predictions of the Circular Law, with eigenvalue density increasing uniformly within the unit disk as matrix size grows. Furthermore, pseudospectral analyses revealed structural complexities and heightened spectral sensitivity within the unit disk—intensifying with larger matrix dimensions—thereby demonstrating that eigenvalues ​​alone may not provide a sufficient picture of system stability. The numerical reliability of the results was confirmed by examining the residual norm, which achieved high levels of precision approaching the machine epsilon limit of $10^{-15}$, thereby ensuring the validity of the proposed scientific conclusions.