- Does diagonalizable imply distinct eigenvalues?
Solution
No. For example, the identity matrix \[ I_2=\begin{pmatrix}1&0\\0&1\end{pmatrix} \] is already diagonal, hence diagonalizable, but it has only one eigenvalue, namely \(1\), with algebraic multiplicity \(2\).
More generally, a diagonalizable matrix may have repeated diagonal entries. Diagonalizability means that the vector space has a basis of eigenvectors; it does not require the corresponding eigenvalues to be distinct.