Eigen value not distinct

Question

If the eigen value of $n\times n$ matrix are not all distinct then does that imply eigen vectors are linearly dependent and hence not diagonalizable?

Answer 1

No, when there are repeated eigenvalues the topics of geometric vs algebraic multiplicity come into play. When the geometric and algebraic multiplicities all equal each other, the matrix is diagonalizable. For example, consider the $n\times n$ identity matrix. It is obviously diagonalizable, and yet it has repeated eigenvalues.

Answer 2

No, consider the identity $n\times n$ matrix.

All its eigenvalues are $1$ but every vector is an eigenvector and it is of course diagonalzible