Eigenvectors corresponding to distinct eigenvalues are linearly independent. This means if λ₁ ≠ λ₂, then X₁ cannot be a scalar multiple of X₂.
∃ a constant K such that \(\overline{X_1} = \overline{KX_2}\), whenever \(\lambda_1 \neq \lambda_2\)
This is false. Eigenvectors for different eigenvalues cannot be scalar multiples of each other.
∃ no constant K such that \(\overline{X_1} = \overline{KX_2}\), whenever \(\lambda_1 \neq \lambda_2\)
This is true. This directly follows from the linear independence of eigenvectors for distinct eigenvalues.
∃ no constant K such that \(\overline{X_1} = \overline{KX_2}\), whenever \(\lambda_1 = \lambda_2\)
This is false. When eigenvalues are equal, eigenvectors may be scalar multiples (though not necessarily).
\(\overline{X_1} = \overline{X_2}\) whenever \(\lambda_1 = \lambda_2\)
This is false. Equal eigenvalues don't imply identical eigenvectors, just that they belong to the same eigenspace.
The correct option is: Option 2