Fact 2 of the diagonalisation note claimed geometric multiplicity never exceeds algebraic multiplicity, without proof. This sub-note proves it, then works a defective matrix in full — one where the inequality is strict, diagonalisation genuinely fails, and no basis, however cleverly chosen, can turn the matrix diagonal.
Learning objectives. By the end of this note you should be able to:
- prove that eigenvectors for distinct eigenvalues are automatically independent;
- prove geometric multiplicity never exceeds algebraic multiplicity;
- construct and verify a concrete defective matrix;
- explain, for the defective example, exactly which basis vector is "missing."
In this lesson
- Eigenvectors for distinct eigenvalues are independent
- Geometric multiplicity never exceeds algebraic multiplicity
- A defective matrix, worked in full
Before you start
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