If your characteristic equation gives

as a "double" eigenvalue and

as a "single" eigenvalue, you

**can't** have two independent eigenvalues corresponding to

. The number of independent eigenvectors corresponding to an eigenvalue (the "geometric multiplicity") cannot be larger than the multiplicity of the eigenvalue (the "algebraic multiplicity").

If

is a double eigenvalue ("algebraic multiplicity" two) with only one independent eigenvector ("geometric multiplicty one), then the matrix is not diagonalizable.