Practice question · Sort into groups
Sort: which properties guarantee diagonalisability, and which do not?
Groups: Guarantees diagonalisability · Does NOT guarantee
- Matrix has a repeated eigenvalue
- Matrix is symmetric
- Matrix is real
- Matrix is invertible
- Enough independent eigenvectors exist
- All eigenvalues are distinct
Hints
- Distinctness and symmetry each guarantee it.
- Invertibility says nothing about the number of independent directions.
Show the answer
Guarantees diagonalisability: All eigenvalues are distinct, Matrix is symmetric, Enough independent eigenvectors exist
Does NOT guarantee: Matrix has a repeated eigenvalue, Matrix is invertible, Matrix is real
Why
Distinct eigenvalues or symmetry guarantee it. Repeated eigenvalues, invertibility, or being real are not sufficient conditions.
Practise Matrix Diagonalisation
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More questions on Matrix Diagonalisation
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