A Theory of Eigenvalue Candidates for 3×3 Matrices Based on Local 2×2 Minors

This paper develops a self-contained algebraic framework for eigenvalue candidates for 3×3 matrices based on the local 2×2 minor equations of A − λI. A unified representation of the parameter-dependent minors is derived, showing that nonprincipal and principal minors produce affine and quadratic candidate equations, respectively. The nonprincipal equations yield explicit minor-ratio candidates, while the roots of each principal-minor equation are precisely the eigenvalues of the corresponding 2×2 principal submatrix. To determine when a local candidate belongs to the spectrum of the full matrix, a local proportionality completion theorem is established. Under the assumptions that the selected local minor vanishes and the corresponding local row segment is nonzero, the determinant of A − λI factors into two completion factors. These factors characterize the linear dependence of the relevant full rows or full columns of the shifted matrix. For principal-minor candidates, a block determinant identity involving the adjugate matrix provides a necessary and sufficient coupling condition. The classical rank-one factorization of the adjugate matrix at rank two is then used to connect local minors with the coordinates of right and algebraic left eigenvectors. This result is included as part of the established adjugate-eigenvector literature and is used here to clarify the detection mechanism of the local candidate equations. Finally, isolated candidate sets and detected-eigenvalue sets generated by the local equations are defined. The local candidate condition is shown to be, in general, neither sufficient nor necessary for membership in the full spectrum. The contribution of this paper lies in organizing the nine local 2×2 minor equations into a unified candidate-generation and validation framework, rather than replacing the characteristic equation.