Let \(A\) be a \(3 \times 3\) matrix with real entries. Prove that, if \(A\) is not similar over \(\mathbb R\) to a triangular matrix, then \(A\) is similar over \(\mathbb C\) to a diagonal matrix.
Solution
1. Failure of real triangularizability means that the characteristic polynomial does not split over \(\mathbb R\).
A linear operator on a finite-dimensional vector space over a field \(F\) is triangularizable over \(F\) if and only if its characteristic polynomial splits into linear factors over \(F\) [@HK71]. Indeed, a triangular matrix has its eigenvalues on the diagonal, so its characteristic polynomial splits. Conversely, if the characteristic polynomial splits, one chooses an eigenvector, passes to the induced operator on the quotient by its span, and argues by induction on the dimension to obtain an invariant flag and hence an upper-triangular basis.
Thus the hypothesis implies that \[ p_A(t)=\det(tI-A) \] does not split completely over \(\mathbb R\).
2. The three complex roots of \(p_A\) are distinct.
The polynomial \(p_A\) is a real monic cubic. Every real polynomial of odd degree has a real root, so write one root as \(\lambda\in\mathbb R\).
Because \(p_A\) does not split over \(\mathbb R\), its remaining two roots are not both real. Since its coefficients are real, every nonreal root occurs together with its complex conjugate. Hence the roots are \[ \lambda,\qquad \mu,\qquad \overline\mu, \] with \(\mu\notin\mathbb R\). These are pairwise distinct: \(\mu\ne\overline\mu\) because \(\mu\) is nonreal, and neither can equal the real number \(\lambda\).
3. Distinct eigenvalues give a complex eigenbasis.
2. Eigenvectors belongingto distinct eigenvalues are linearly independent [@HK71]. Choosing onenonzero eigenvector for each of\(\lambda,\mu,\overline\mu\) therefore gives three linearly independent vectorsin the three-dimensional complex space \(\mathbb C^3\). They form a basis inwhich \(A\) has diagonal matrix\[ \operatorname{diag}(\lambda,\mu,\overline\mu). \]Hence \(A\) is similar over \(\mathbb C\) to a diagonal matrix.