Rank of a diagonalizable matrix?

Question

What can be said about the rank of a diagonalizable matrix?

Answer 1

Let $e_i$ be the $n \times 1$ vector with a $1$ in the $i$-th component and $0$'s everywhere else.

For any $0 \le k \le n$, the $n \times n$ matrix $A = \displaystyle\sum_{i = 1}^{k}e_ie_i^T$ is diagonalizable has rank $k$.

Hence, a matrix being diagonalizable tells you nothing about its rank.

Answer 2

The rank of a diagonalizable matrix is the same as the rank of its diagonalization. The latter is easy to compute by looking at its entries, since the rank of a diagonalized matrix is simply the number of nonzero entries.

Answer 3

The rank is the number of non-zero eigenvalues.