Geometry and algebra of linear maps, vectors, and matrices
Diagonalization rewrites a matrix in its own most natural coordinate system, the one built from its eigenvectors. In that system the matrix is diagonal: it does nothing but scale each eigen-axis by its eigenvalue. A tangled transformation becomes a simple one.
Here P has the eigenvectors as its columns and D is diagonal with the eigenvalues. Read the product right-to-left as a three-step recipe: P⁻¹ rotates into eigen-coordinates, D scales each axis, and P rotates back. A messy transformation, expressed as a pure stretch between two changes of view.
Diagonalization makes matrix powers almost free. Because the middle P⁻¹P pairs cancel, Aᵏ = P Dᵏ P⁻¹, and raising a diagonal matrix to a power just raises each diagonal entry to that power. No repeated matrix multiplication needed.