alleryStart learning →For a SYMMETRIC real matrix $A$, all eigenvalues are
Acomplex
Breal
Cpurely imaginary
Dpositive
Answer & Solution
Correct answer: B. real
1. SPECTRAL THEOREM (for real symmetric matrices): all eigenvalues are REAL.
2. Moreover: eigenvectors corresponding to DIFFERENT eigenvalues are ORTHOGONAL.
3. Conclusion: a real symmetric matrix is ALWAYS diagonalisable by an ORTHOGONAL matrix: $A = Q D Q^T$.
4. POSITIVE eigenvalues (option D) characterise POSITIVE-DEFINITE symmetric matrices — a stronger condition.
5. Options A, C contradict the spectral theorem.
_Source: Sergei Treil, "Linear Algebra Done Wrong", §5.2 (Spectral theorem for symmetric matrices)._
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