Diagonalizability and linear independence of columns

Question

Suppose $\Phi_{n\times n}$ is a matrix whose columns are $n$ linearly independent vectors of $\mathbb{R}^n$. So, $\Phi$ has an inverse.

I want to know whether $\Phi$ is always diagonalizable or not.

Answer 1

No. Consider $\Phi=\begin{pmatrix}1&1\\0&1\end{pmatrix}$ which is invertible but not diagonalizable.

Moreover, $\Psi=\begin{pmatrix}1&0\\0&0\end{pmatrix}$ is diagonalizable (since it is a diagonal matrix), but not invertible hence the columns are linear dependent.