2.2.6—Eigen terminology

For a square matrix, the sum of eigenvalues equals the trace and the product of eigenvalues equals the determinant, counting algebraic multiplicity.

These invariants quickly check a characteristic polynomial or numerical answer, but they do not determine eigenvectors or guarantee diagonalisation.

For a 2×2 matrix with trace 5 and determinant 6, possible eigenvalues 2 and 3 pass both checks; an alleged pair 1 and 5 fails the determinant check.

Matching trace and determinant is necessary, not sufficient, for a complete eigen-analysis; the eigenspaces still need to be solved.