This closing note of Topic 4 draws out the payoff of the spectral theorem for a question the unit has carried since LA-I's diagonalisation topic: can a symmetric matrix ever be defective? Theorem 2 of the diagonalisation note stated the general diagonalisability criterion (algebraic multiplicity = geometric multiplicity, for every eigenvalue) without a way to guarantee it held for any particular matrix class. This note proves it always holds for symmetric matrices, closing the question completely.
Learning objectives. By the end of this note you should be able to:
- prove that a real symmetric matrix's algebraic and geometric multiplicities always coincide;
- explain precisely how the spectral theorem rules out defectiveness;
- restate LA-I's diagonalisability criterion in light of this stronger, symmetric-specific guarantee;
- identify which of the earlier unit's worked defective examples this consequence explains.
In this lesson
- No defective symmetric matrices
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