Complex Eigenvalues, Orthogonality, and QR Factorization: Analytical Proofs and Numerical Verification

This research provides a rigorous investigation into complex eigenvalues and eigenvectors, inner products, and orthogonality in the context of Hermitian and normal matrices. The work includes both analytical proofs and numerical verification to ensure reproducibility and clarity.The methodology begins with a formal examination of Hermitian matrices, establishing that all eigenvalues are real and eigenvectors corresponding to distinct eigenvalues are orthogonal. These results are then validated through computational experiments using Python with the NumPy and SciPy libraries.Key computational techniques include:Eigenvalue and Eigenvector Computation: Generation of random Hermitian matrices and computation of their eigenvalues and eigenvectors. Orthogonality is verified numerically using absolute inner products, visualized through heatmaps.Gram–Schmidt Orthonormalization: Construction of orthonormal bases from linearly independent vectors. Reconstruction error and orthonormality error are computed and compared to SciPy’s QR decomposition to ensure numerical accuracy.QR Factorization and Least-Squares Solutions: QR decomposition is applied to overdetermined systems to solve least-squares problems. Discrepancies between QR-based solutions and NumPy’s lstsq method are quantified to confirm high accuracy (~10⁻¹⁶), demonstrating numerical stability and machine-precision correctness.All four figures referenced in this work are generated programmatically and saved in the eigenvalues/ folder, allowing full reproduction of results:Figure 1: Eigenvector orthogonality heatmapFigure 2: Gram–Schmidt vs SciPy QR reconstruction errorsFigure 3: Least-squares solution comparisonFigure 4: Gram–Schmidt vectors orthonormality heatmapNo human or animal data were used, and no ethical approval was required, as all computations are fully synthetic.This study provides a computationally verified, reproducible framework for exploring key concepts in numerical linear algebra, and it is suitable as both a teaching resource and a reference for researchers implementing high-precision matrix computations.