When reading eigenvector of a matrix, there is a formula:
$AP = PD$
where in $P$, each column is A's eigenvector and $D$ is diagonal matrix with diagonal element being A's eigen values.
Now coming the question:
Is matrix P always invertible? because I often see equation $A = PDP^{-1}$ here and there, but don't know how to proof.
Best Answer
If $A$ is diagonalizable, then yes. In some (all? most?) texts the very definition of an $n\times n$ matrix $A$ being diagonalizable over a field $F$ (let's assume $\mathbb{R}$) is that there exists a basis of $\mathbb{R}^n$ made from the eigenvectors of $A$.
The columns of $P$ are exactly these eigenvectors, and them being a basis implies their linear independence. Hence $P$ is an invertible matrix.