Why do diagnolizable matrices have to be invertible? My professor gave us this definition of a diagnolizable matrix. A matrix $A$ is diagnolizable if it's invertible and 
$$(Ax)_{\mathcal{B}} = Dx_{\mathcal{B}}$$
for some diagonal matrix $D$, basis $\mathcal{B}$ and all vectors $x$.
But there are non invertible matrices that have "diagonal equivalents" in other basis. For example, 
$\begin{bmatrix}2 & -1\\2 & -1\end{bmatrix}$ is diagnolizable to $\begin{bmatrix}1 & 0\\0 & 0\end{bmatrix}$ even though its not invertible.
So why the restriction on the definition?
 A: Diagonalizable matrices need not be invertible. For instance, the matrix $$A = \begin{bmatrix}
1 & 1\\
1 & 1
\end{bmatrix} = 
\begin{bmatrix}
-1/\sqrt2 & 1/\sqrt2\\
1/\sqrt2 & 1/\sqrt2\\
\end{bmatrix}
\begin{bmatrix}
0 & 0\\
0 & 2\\
\end{bmatrix}
\begin{bmatrix}
-1/\sqrt2 & 1/\sqrt2\\
1/\sqrt2 & 1/\sqrt2\\
\end{bmatrix}$$
The converse also need not hold true, i.e., invertible matrices need not be diagonalizable. For instance,
$$A = \begin{bmatrix}1 & 1\\ 0 & 1 \end{bmatrix}$$ is invertible but not diagonalizable.
Diagonalizability and Invertibility do not relate to each other, in general.
A: First of all, my apologies for my mixup in your other post. Hope this will help clarify the issue:, by solving :$Ax=x$ and $(A-I)x=x$ for $x$.
A non-invertible matrix can be diagonalizable. A necessary and sufficient condition for a matrix $A$ to be invertible is that the matrix must have a basis of eigenvectors, so that the dimension of the eigenspace of the matrix must be $\geq n$ , for $A$ an $n \times n$ matrix. Consider your matrix:
$\begin{bmatrix}2 & -1\\2 & -1\end{bmatrix}$
Let's compute its eigenspace. first, the eigenvalues. Set:
$Det\begin{bmatrix}2 -\lambda & -1\\2 & -1-\lambda\end{bmatrix}=0$
The characteristic polynomial is : $\lambda^2-\lambda=\lambda(\lambda -1):=0$
Then there are two eigenvalues; $\lambda=1,0$. This guarantees that you have two linearly-independent eigenvectors. Maybe you want to compute the eigenspace associated with each of the eigenvalues.
