1. The Goal: What Diagonalization Means
Plain-language goal: We want to write a square matrix $A$ as a product
$$A = P\,D\,P^{-1}$$
where $P$ is an invertible matrix whose columns are eigenvectors, and $D$ is a diagonal matrix whose diagonal entries are the corresponding eigenvalues.
Why does this matter?
Think of $A$ as a linear transformation. The factorisation $A = PDP^{-1}$ says:
- Change coordinates — multiply by $P^{-1}$ to move into the eigenvector basis.
- Scale each axis independently — multiply by $D$, which simply multiplies each coordinate by its eigenvalue.
- Change back — multiply by $P$ to return to the original coordinates.
Change-of-Basis Reading
Recall from our earlier work on bases that the same linear map can look very different in different coordinate systems. Diagonalization is the best possible change of basis: in the eigenvector basis, the transformation is nothing more than axis-wise scaling. Every complicated rotation-and-stretch that $A$ performs is revealed as simple stretching along special directions.
The Payoff — Why Learn the Machinery?
- Matrix powers: $A^k = PD^kP^{-1}$, and raising a diagonal matrix to a power is trivial — just raise each diagonal entry to that power.
- Decoupling: Systems of equations that are tangled together become independent one-variable problems in the eigenvector basis.
- Insight: The eigenvalues on the diagonal tell you the long-run behaviour of the system at a glance — which directions grow, which shrink, which stay fixed.
Interactive Visualisation — Diagonalization as Change of Basis + Scaling
Interactive visualization — available in the full lesson
The blue square is the unit square in standard coordinates. The green parallelogram shows the eigenvector basis. The red shape shows the result of applying $A$. Drag the slider to interpolate the three steps of diagonalization.