Practice question · Multiple choice
Not every matrix is diagonalizable. What is actually missing when one is not?
Hints
- Diagonalising means finding a basis of eigenvectors. Ask what could go wrong with that hunt.
- A shear matrix has one eigenvalue and only one independent eigenvector. What does that leave you short of?
Show the answer
A. Enough independent eigenvectors to form a basis
Why
Diagonalising is choosing a basis of eigenvectors, so the failure is not having enough of them, a shear has a repeated eigenvalue and a single direction, leaving the basis one vector short. Jordan normal form is what you settle for: as close to diagonal as such a matrix can be pushed.
Practise Diagonalization
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