Diagonalization Checker Calculator

FAQs

1. What does “diagonalizable” mean? A square matrix is diagonalizable if there exists an invertible matrix P such that P⁻¹AP is diagonal.
2. Why does symmetry guarantee diagonalization over ℝ? By the spectral theorem, every real symmetric matrix has an orthonormal eigenbasis and real eigenvalues.
3. How are eigenvalues computed here? Using a shifted QR iteration to approximate the real Schur form, then extracting 1×1 and 2×2 blocks.
4. What if my matrix has complex eigenvalues? It cannot be diagonalized over ℝ but may be diagonalizable over ℂ. The tool flags such cases and provides a heuristic over ℂ.
5. What causes “inconclusive” over ℂ? Repeated complex eigenvalues require complex-nullspace checks, which this page does not compute. Use a CAS for a definitive answer.
6. Why provide a tolerance? Numerical algorithms require thresholds to decide when values are “effectively zero,” influencing rank and multiplicity judgments.
7. What sizes work best? The QR iteration here targets small matrices (2–6). Larger or ill-conditioned problems are better handled with specialized software.