Is there any connection between a matrix being invertible and being diagonalizable?

linear-algebra inverse diagonalization

One does not imply the other.

This matrix is invertible and not diagonalizable:

$\begin{pmatrix} 1 & 1 \\ 0 & 1\\ \end{pmatrix}$

This matrix is diagonalizable (in fact it is already a diagonal matrix) but not invertible:

$\begin{pmatrix} 0 & 0 \\ 0 & 0 \\ \end{pmatrix}$

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