Diagonalization
Diagonalization is a more specific form of Similar Matrices, basically, diagonalization is the process of finding a matrix $D$ that is similar to $A$ and is a diagonal matrix. As well as finding the matrix $P$. This makes it easier to calculate the $k^{th}$ time the matrix $A$ is applied to a vector $v$.
In diagonalization, the matrices $P$ and $D$ are calculated using the [notes/Eigenvalues|Eigenvalues] and their coressponding [notes/Eigenvectors|Eigenvectors].
$$ D = \begin{bmatrix} \lambda_{0} & 0 & 0 & 0 & … \\0 & \lambda_{1} & 0 & 0 & … \\0 & 0 & \lambda_{2} & 0 & … \\0 & 0 & 0 & \lambda_{3} & … \\\end{bmatrix} $$
$$ P = \begin{bmatrix} w_0 & w_{2} & w_{3} & w_{4} & …\\\end{bmatrix} $$
Since one of the conditions for diagonability is that there are $n$ linearly independent eigenvectors, $P$ must be invertible.