<![CDATA[This paper investigates the spectral properties of the symmetric circulant matrix ??(?,?)=circ (?,?,0,…,0,?), where ?,?∈ℝ and ?≥3. While the eigenvalue structure of general circulant matrices is well understood, explicit and unified characterizations for specific structured subclasses remain of interest. In this work, we derive closed-form expressions for the eigenvalues and provide a complete characterization of the positive definiteness of this matrix family, explicitly highlighting the role of the parity of ?. In addition, we obtain a compact determinant formula using Chebyshev polynomials, yielding an analytically tractable condition for singularity. The results establish a direct connection between circulant matrix theory and classical trigonometric polynomial identities, providing a unified framework that links spectral properties, determinant structure, and parity effects. These findings extend existing formulations by providing explicit, structurally transparent results for this class of symmetric circulant matrices.]]>
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