Diagonalisable matrix
diagonal for some invertible , equivalently a basis of eigenvectors exists; this holds exactly when for every eigenvalue .
Statement
For an matirx acting on or , the following are equivalent:
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There exists a basis of consisting of eigenvectors of . i.e. we have where
for some eigenvalue .
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is diagonalisable, i.e. there exists an invertible matrix such that
where is a diagonal matrix, with the eigenvalues of on the diagonal:
Criteria for diagonalisability
An matrix with distinct eigenvalues is diagonalisable: the corresponding eigenvectors are linearly independent, hence form a basis of or . This criterion is sufficient but not necessary.
Necessary and sufficient: for every eigenvalue ,
Indeed, taking bases of the eigenspaces of the distinct eigenvalues , the union is linearly independent with
so it is a basis of or exactly when no defect remains.
Related
Stated in
- Definition 4.12 (Diagonalisable Matrix)§4.3 Diagonolisation and Similarity
- Proposition 4.11§4.3 Diagonolisation and Similarity