<![CDATA[The eigenvalue problem constitutes one of the fundamental topics in mathematical physics, particularly in quantum mechanics, operator theory, and dynamical system analysis, since physical observables are no longer represented directly by scalar quantities but rather by linear operators expressed in matrix form. In this context, symmetric and skew-symmetric matrices become important objects of study because they exhibit distinct spectral properties, which have direct implications for the interpretation of eigenvalues and eigenvectors in physical systems. This study investigates the eigenvalue problem for symmetric and skew-symmetric matrices through a basis transformation approach and the diagonalization process using a unitary matrix. Symmetric matrices possess a real eigenvalue spectrum and orthonormal eigenvectors, and thus can be diagonalized directly by means of a unitary transformation, as expressed by the relation D=UBU. In contrast, have purely imaginary eigenvalues occurring in conjugate pairs. This study is conducted through both analytical and computational approaches by employing eigenvalue computation algorithms and orthonormalization procedures to construct a unitary basis. The results demonstrate that unitary basis transformations are capable of simplifying the operator representation into a diagonal form, thereby facilitating the analysis of physical systems for both symmetric and skew-symmetric matrices. Accordingly, this study emphasizes that unitary transformations and diagonalization constitute essential mathematical tools for understanding the spectral structure of operators in a wide range of physical problems.]]>
