A linear transformation becomes especially simple when we choose a basis made entirely of eigenvectors. In that basis, each coordinate direction is merely scaled independently.

A square matrix $A$ is diagonalizable when there is an invertible matrix $P$ and a diagonal matrix $D$ such that

$$A=PDP^{-1}.$$

Where $P$ and $D$ come from

The columns of $P$ are linearly independent eigenvectors of $A$. The corresponding eigenvalues appear on the diagonal of $D$ in the same order.

If

$$A\mathbf v_i=\lambda_i\mathbf v_i,$$

then

$$P=\begin{pmatrix}\mathbf v_1&\cdots&\mathbf v_n\end{pmatrix},$$

$$D=\begin{pmatrix} \lambda_1&&0\ &\ddots&\ 0&&\lambda_n \end{pmatrix}.$$

A change of coordinates

The equation

$$A=PDP^{-1}$$

can be read from right to left:

1. $P^{-1}$ converts a vector into eigenvector coordinates;
2. $D$ scales each eigenvector coordinate independently;
3. $P$ converts back to the original coordinates.

So diagonalization does not change the transformation. It finds coordinates in which its action is simplest.

When diagonalization is possible

An $n\times n$ matrix is diagonalizable exactly when it has $n$ linearly independent eigenvectors. Distinct eigenvalues automatically provide independent eigenvectors, so a matrix with $n$ distinct eigenvalues is diagonalizable.

Repeated eigenvalues require more care: there must still be enough independent vectors across the eigenspaces.

Why diagonal form is useful

Powers become easy to compute:

$$A^k=PD^kP^{-1},$$

and

$$D^k=\operatorname{diag}(\lambda_1^k,\ldots,\lambda_n^k).$$

This makes diagonalization useful for repeated transformations, recurrence relations and differential equations.

Visual synthesis: eigenvectors as a change of basis